{"id":162,"date":"2019-03-11T09:47:34","date_gmt":"2019-03-11T09:47:34","guid":{"rendered":"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=162"},"modified":"2019-03-11T10:25:17","modified_gmt":"2019-03-11T10:25:17","slug":"introduction-to-probability","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/chapter\/introduction-to-probability\/","title":{"rendered":"Introduction to Probability"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/u5sFDxoIvvo\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div>\r\n\r\n\u00a0 \u00a0\u00a0<strong>Learning Objectives<\/strong>\r\n<ul>\r\n \t<li><strong>Introduction<\/strong><\/li>\r\n \t<li><strong>Terminology<\/strong><\/li>\r\n \t<li><strong>Types of Events<\/strong><\/li>\r\n \t<li><strong>Probability Approach<\/strong><\/li>\r\n \t<li><strong>Axioms of Probability<\/strong><\/li>\r\n \t<li><strong>Permutations<\/strong><\/li>\r\n \t<li><strong>Combination<\/strong><\/li>\r\n \t<li><strong>Summary<\/strong><\/li>\r\n \t<li><strong>Suggested Readings<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0\u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">The objective of this module is to introduce key concepts that help to understand <\/span>probability<span style=\"text-align: justify;font-size: 1em\"> problem in simple words and different approaches to find <\/span>numerical<span style=\"text-align: justify;font-size: 1em\"> measure of probability for different probability problem. Examples and some images are used for <\/span>better<span style=\"text-align: justify;font-size: 1em\"> explanation of this topic in an easy manner.<\/span><\/p>\r\n\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>2.<\/strong>\u00a0\u00a0 <strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Probability measures the uncertainty of an event in an experiment. Theory of probability is an important branch of mathematics and statistics that provides numerical measure of the probability. Probability is also used in our daily life. Likely, possible, high chance of use of like words in day-to-day life to express uncertainty of happening an event. For example: possibly, it will rain today and there is a high chance to get my dream job in next two months.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">According to Ya-Lin-Chou \u201cProbability is the science of decision making with calculated risks in the face of uncertainty\u201d.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Probability phenomenon is mostly observed in business, economics, social science, daily life and actuaries. Probability has many application in economics, business and actuaries because there are terminologies such as risk, taxes, profit, loss etc. These terminologies express uncertainty of an event under an experiment. Suppose an investor invest money in money market. There are chances that either investor will get more money as profit of investment or will face loss in the investment. This shows uncertainty, probability terminology can be used here.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A real life exercise of number plate of Victorian is shown in Figure 1, you will be able to solve these types of exercises after completing this module on probability.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-167\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78.png\" alt=\"\" width=\"430\" height=\"302\" \/>\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">3<\/strong><span style=\"text-align: initial;font-size: 1em\">.<\/span><strong style=\"text-align: initial;font-size: 1em\"> Terminology<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There is a need to be aware <\/span>about<span style=\"text-align: initial;font-size: 1em\"> some basic concepts of probability before going to <\/span>numerical<span style=\"text-align: initial;font-size: 1em\"> measure of probability. In this section, <\/span>focus<span style=\"text-align: initial;font-size: 1em\"> will be on key concepts to understand a probability problem in words and convert into numerical form to solve it by using <\/span>different<span style=\"text-align: initial;font-size: 1em\"> approach of probability according to the features of the problems.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nKey concepts are given as:\r\n\r\n&nbsp;\r\n\r\n<strong>3.1 Random Experiment<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Before conducting an experiment there is no possibility to get outcome of any problem. Experiment is a process to observe an uncertainty in the problem.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cRandom experiment is an experiment in which outcome of an trial or event under some identical circumstances are not unique but it will be among all the possible outcomes.\u201d<\/p>\r\n&nbsp;\r\n\r\nList of some examples of random experiment is given below:\r\n<ul>\r\n \t<li>Tossing a coin.<\/li>\r\n \t<li>Rolling a dice.<\/li>\r\n \t<li>Observe risk<span style=\"text-align: initial;font-size: 1em\"> of an investor.<\/span><\/li>\r\n \t<li>Selecting a card from a deck of card.<\/li>\r\n \t<li>Selecting a ball from a group of balls.<\/li>\r\n \t<li>Arrangement<span style=\"text-align: initial;font-size: 1em\"> of persons in a queue.<\/span><\/li>\r\n \t<li>Observe the number of Xiaomi android phones sold by Amazon in 2017.<\/li>\r\n \t<li>Selecting a student from a classroom.<\/li>\r\n \t<li>Observe the number of wickets in a cricket match.<\/li>\r\n \t<li>Observe product<span style=\"text-align: initial;font-size: 1em\"> of a process in a manufacturing company. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Observe a bulb is either fuse or not.<\/span><\/li>\r\n \t<li>Observe a person who speaks either truth or lie.<\/li>\r\n \t<li>Observe the number of car<span style=\"text-align: initial;font-size: 1em\"> accident in <\/span>particular<span style=\"text-align: initial;font-size: 1em\"> place. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Observe the number of birth in India in <\/span>year<span style=\"text-align: initial;font-size: 1em\"> 2017.<\/span><\/li>\r\n \t<li>Observe the share market prices.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 3.2 Outcome<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">After conduct of a random experiment, now to observe the uncertainty of a problem, there will be some results or outcome of that experiment. Outcome gives us an idea about all possible results of that experiment.<\/p>\r\n&nbsp;\r\n\r\n\u201cOutcome is a result of a random experiment\u201d.\r\n\r\nList of some outcomes of random experiment is given as:\r\n<ul>\r\n \t<li>Outcome of tossing a coin:\u00a0 head (H) or tail (T).<\/li>\r\n \t<li>Outcome of rolling a dice: 1\/ 2\/ 3\/ 4\/ 5\/ 6.<\/li>\r\n \t<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of selecting a ball from a group of different colors balls like blue, red, white: blue\/ red\/ white.<\/span><\/li>\r\n \t<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of <\/span>risk<span style=\"text-align: initial;font-size: 1em\"> of an investor: high risk\/ neutral risk\/ low risk\/ no risk.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Outcome of selecting a card from a deck of card: 1\/ 2\/ 3\/\u2026. \/10\/ J\/ Q\/ K\/ A\/ club\/ spade\/ diamond\/ heart\/ red\/ black.<\/li>\r\n \t<li style=\"text-align: justify\">Outcome<span style=\"text-align: initial;font-size: 1em\"> of <\/span>working<span style=\"text-align: initial;font-size: 1em\"> condition of a bulb: Fuse\/not fuse. <\/span>Outcome<span style=\"text-align: initial;font-size: 1em\"> of a person speaking tendency; truth\/ lie.<\/span><\/li>\r\n \t<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of observing car accident in a state: 1, 2, \u2026\u2026<\/span><\/li>\r\n \t<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of product quality in a manufacturing company: defective\/ non-defective.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a03.3 Sample space:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">After identifying the result of the experiment, there is a need to arrange all possible outcome. In basic mathematics, the definition of <\/span><strong style=\"text-align: initial;font-size: 1em\">set<\/strong><span style=\"text-align: initial;font-size: 1em\"> was described. Set is a group of well-defined objects <\/span>that related<span style=\"text-align: initial;font-size: 1em\"> to each other under <\/span>same<span style=\"text-align: initial;font-size: 1em\"> consideration. Here, <\/span>result<span style=\"text-align: initial;font-size: 1em\"> of an experiment is well defined different result under <\/span>same<span style=\"text-align: initial;font-size: 1em\"> consideration, so set is used as sample space to arrange all possible outcomes.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cSample space is set of all possible outcomes of an experiment\u201d.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">List of some examples of sample space <\/span>are<span style=\"text-align: initial;font-size: 1em\">:<\/span><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Sample space for tossing a coin: {H, T}.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Sample space for rolling a dice: {1, 2, 3, 4, 5, 6}.<\/li>\r\n \t<li style=\"text-align: justify\">Sample space for selecting a card from a deck of card: {1, 2, \u2026, 10, A, K, Q, J}.<\/li>\r\n \t<li style=\"text-align: justify\">Sample space for selecting a ball from a group of balls of different<span style=\"text-align: initial;font-size: 1em\"> color like blue, green and red: {Blue ball, red ball, green ball}.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Sample space of observe risk<span style=\"text-align: initial;font-size: 1em\"> of an investor: {high, low, neutral, no}.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Sample space of product that produces by a manufacturing company: {defective, non<span style=\"text-align: initial;font-size: 1em\">-defective}.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Sample space of observe a bulb: {fuse, non-fuse}.<\/li>\r\n \t<li style=\"text-align: justify\">Sample space of observing person talk: {truth, lie}.<\/li>\r\n \t<li style=\"text-align: justify\">Sample space for observe car accident in a state: {1, 2, 3\u2026}.<\/li>\r\n \t<li style=\"text-align: justify\">Sample space for observe<span style=\"text-align: initial;font-size: 1em\"> the number of wickets in a cricket match: {1, 2, \u2026, 22}.<\/span><\/li>\r\n<\/ul>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 3.4 Event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If, one is interested in particular results among all possible results of a random experiment, these particular results can be obtained in terms of events.<\/p>\r\n&nbsp;\r\n\r\n\u201cEvent is a particular performance of a random experiment\u201d.\r\n\r\nList of some examples of event:\r\n<ul>\r\n \t<li>Suppose a random experiment is tossing a coin two times.<\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 Sample space<\/strong>: {HH, HT, TH, TT}\r\n\r\n<strong>Possible event<\/strong>: getting head both times, getting tail both times, getting one head and tail of a coin.\r\n<ul>\r\n \t<li>Suppose rolling a die two times is a random Experiment.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Sample space<\/strong><span style=\"text-align: initial;font-size: 1em\">: {(1,1), (1,2),\u2026\u2026\u2026,(1,6), (2,1), (2,2), \u2026\u2026..,(2,6), (3,1), (3,2),\u2026\u2026.,(3,6), (4,1),\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">(4,2), \u2026\u2026..,(4,6), (5,1), (5,2),\u2026\u2026.,(5,6), (6,1), (6,2),\u2026..,(6,6)}.<\/span><\/p>\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Possible event<\/strong><span style=\"text-align: initial;font-size: 1em\">: Getting <\/span>same<span style=\"text-align: initial;font-size: 1em\"> number both <\/span>times ,<span style=\"text-align: initial;font-size: 1em\"> getting <\/span>odd<span style=\"text-align: initial;font-size: 1em\"> number at <\/span>first<span style=\"text-align: initial;font-size: 1em\"> time and even number at <\/span>second<span style=\"text-align: initial;font-size: 1em\"> time, getting <\/span>odd<span style=\"text-align: initial;font-size: 1em\"> number at first time, getting <\/span>sum<span style=\"text-align: initial;font-size: 1em\"> of both is <\/span>even ,<span style=\"text-align: initial;font-size: 1em\"> getting <\/span>sum<span style=\"text-align: initial;font-size: 1em\"> of both <\/span>time<span style=\"text-align: initial;font-size: 1em\"> is <\/span>odd ,<span style=\"text-align: initial;font-size: 1em\"> many more event may occur.<\/span><\/p>\r\n\r\n<div>\r\n<ul>\r\n \t<li>Selecting a card from a deck of cards is an random experiment.<\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 Sample space<\/strong>: {1, 2, \u2026\u2026,10, A, K, Q, J}.\r\n<p style=\"text-align: justify\"><strong>Possible event<\/strong>: Getting a spade, getting a club, getting a heart , getting a diamond, getting a face card, getting a king, getting a queen , many more event may occur respect to sample space of random experiment.<\/p>\r\n&nbsp;\r\n\r\n<strong>4<\/strong><strong>.<\/strong><strong> Types of Events<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">There are many types of event according to the happening of event and outcome. In this section, types of event are defined.<\/p>\r\n&nbsp;\r\n\r\n<strong>4.1 Simple event:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose a particular event happens among all possible events. Result of particular event is at most one, then event is called simple event.<\/p>\r\n&nbsp;\r\n\r\n\u201cAn event that shows exactly one outcome of a random experiment is a simple event\u201d.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose a quality analyst analysis two processes\u2019 product in a random experiment and classify the process product as defective product and non-defective product.<\/p>\r\n&nbsp;\r\n\r\nSample space: {nn, nd, dn, dd}\r\n\r\n&nbsp;\r\n\r\nwhere d is for defective product and n is for non-defective product.\r\n\r\n&nbsp;\r\n\r\nLet E is an event.\r\n\r\n&nbsp;\r\n\r\nEvent E : neither product is defective.\r\n\r\n&nbsp;\r\n\r\nE ={nn}; as E has only one outcome, E is a simple event.\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Example<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n\u00a0 \u00a0 Tossing a coin two times in a random experiment.\r\n\r\n&nbsp;\r\n\r\nSample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail\r\n\r\n&nbsp;\r\n\r\nLet E be an event: getting head both time.\r\n\r\n&nbsp;\r\n\r\nE = {HH}; E has only one outcome, E is a simple event.\r\n\r\n&nbsp;\r\n\r\n<strong>4.2 Compound event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose a particular event happens among all possible events. Result of particular event is at least one outcome that event is called a compound event.<\/p>\r\n&nbsp;\r\n\r\n\u201cA event shows more than one outcome of a random experiment is a compound event\u201d.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose a quality analyst analysis two processes\u2019 product quality in an random experiment and classify the process product as defective product and non-defective product.<\/p>\r\n&nbsp;\r\n\r\nSample space : {nn, nd, dn, dd}\r\n\r\n&nbsp;\r\n\r\nwhere: d is for defective product and n is for non-defective product.\r\n\r\n&nbsp;\r\n\r\nLet E is an event.\r\n\r\n&nbsp;\r\n\r\nEvent E : at least one product is defective.\r\n\r\n&nbsp;\r\n\r\nE: {nd, dn, dd}; E has more than one outcome. So, E is a compound event.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n\r\nTossing a coin two times is a random experiment.\r\n\r\n&nbsp;\r\n\r\nSample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail\r\n\r\n&nbsp;\r\n\r\nLet E be an event getting head at least one time.\r\n\r\n&nbsp;\r\n\r\nE: {HH, HT, TH}; E has more than one outcome. So, E is a compound event.\r\n\r\n&nbsp;\r\n\r\n<strong>4.3 Exhaustive event<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">All possible results of an experiment <\/span>is<span style=\"text-align: initial;font-size: 1em\"> termed as an exhaustive event.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u201cExhaustive event is the total outcome of a random experiment\u201d.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">List of some examples are:<\/span>\r\n<ul>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Tossing of one coin is a random experiment, exhaustive cases are 2. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Throwing a dice, exhaustive cases are 6.<\/span><\/li>\r\n \t<li>Throwing two dices, exhaustive cases are 62.<\/li>\r\n \t<li>Tossing two coins, exhaustive cases are 22.<\/li>\r\n \t<li>Selecting a card, exhaustive cases are 52.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 4.4 Dependent event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Two particular events E and F occurred in a trail. Result of one event are affecting the result of second event. Event may be either E or F, these events will be dependent event.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cEvent is said to be dependent event if happening of one event in one trail is affected by happening of similar event in next trail\u201d.<\/p>\r\n&nbsp;\r\n\r\n<strong>Examples:<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Drawing a card from a pack of card followed by one after another draw is a random experiment. Let us consider E is an event of drawing black card. If first drawn card is not replaced in the deck of card while second card is drawn, then the second time drawn card will be dependent on the first drawn card. After this total number of cards are decreasing followed by one after another draw. So, E is a dependent event.<\/li>\r\n \t<li style=\"text-align: justify\">Let us consider a bag with five marbles, four are red and one is blue. If a marble is selected at random from the bag and not replaced again into the bag, the chance of occurrence of the blue ball will keep increasing with each consecutive draw.<\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 4.5 Favorable event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Outcomes of a particular event are favorable to that events. These outcomes of events are considered as favorable event.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cFavorable event is the number of outcome favor to an event\u201d.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Examples<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\">Tossing\u00a0 two\u00a0 coins\u00a0 is\u00a0 a<span style=\"text-align: initial;font-size: 1em\">\u00a0 <\/span>random\u00a0 experiment\u00a0 and\u00a0 getting\u00a0 head\u00a0 on\u00a0 both\u00a0 coins\u00a0 is\u00a0 an\u00a0 event<span style=\"text-align: initial;font-size: 1em\">.\u00a0<\/span>Exhaustive<span style=\"text-align: initial;font-size: 1em\"> event is 22 = 4 and <\/span>favorable<span style=\"text-align: initial;font-size: 1em\"> event is 1 (i.e. head and head).<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Throwing two dice is an random<span style=\"text-align: initial;font-size: 1em\"> experiment and getting <\/span>same<span style=\"text-align: initial;font-size: 1em\"> number on both <\/span>side<span style=\"text-align: initial;font-size: 1em\"> is an event, <\/span>favorable<span style=\"text-align: initial;font-size: 1em\"> event is 6 {(1,1), (2,2), (3,3), (4,4), ( 5,5), (6,6)}.<\/span><\/li>\r\n<\/ul>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a04.6 Mutually exclusive event<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u201cEvents can\u2019t occur simultaneously in <\/span>same<span style=\"text-align: initial;font-size: 1em\"> trial are mutually exclusive events\u201d.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Suppose A and B are two events. If A and B are <\/span>mutually<span style=\"text-align: initial;font-size: 1em\"> exclusive event, their <\/span>venn<span style=\"text-align: initial;font-size: 1em\"> diagram is given\u00a0in Figure 2.<\/span><span style=\"text-align: initial;font-size: 1em\"><img class=\"aligncenter size-full wp-image-168\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-79.png\" alt=\"\" width=\"237\" height=\"145\" \/><\/span>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\">Figure 2<\/p>\r\n&nbsp;\r\n\r\n<strong>Examples<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Head and tail are two outcomes of random experiment of a coin. Suppose A and B are two events of getting head and getting tail respectively. Both event cannot occur in a single trail, so A and B are mutually exclusive event.<\/li>\r\n \t<li style=\"text-align: justify\">1, 2, 3, 4, 5, 6 are the possible six outcome<span style=\"font-size: 1em\"> of <\/span>random<span style=\"font-size: 1em\"> experiment in throwing <\/span>a dice<span style=\"font-size: 1em\">. All of these outcomes cannot occur in a single trail, so they are all mutually exclusive event.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">A switch cannot be on\/ off at same<span style=\"text-align: initial;font-size: 1em\"> time. Switching a switch button is a random experiment that has two possible outcomes on and off that switch is on while the second one shows switch is off. Both <\/span>event<span style=\"text-align: initial;font-size: 1em\"> cannot occur at <\/span>same<span style=\"text-align: initial;font-size: 1em\"> time, so A and B are <\/span>mutually<span style=\"text-align: initial;font-size: 1em\"> exclusive event.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>\u00a0 \u00a0 4.7 Independent event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Two particular event E and F are occurred in a trial. Result of one event is not affecting the result of second event. Event may be either E or F, so these events will be independent event.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cEvent is said to be independent event if happening of one event in one trail is not affected by happening of similar event in next trial\u201d.<\/p>\r\n&nbsp;\r\n\r\n<strong>Examples<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Throwing a dice is a random experiment, if one get six in first throw as well as in second throw. One can see that both throws of a dice are not affecting the outcome of each other, so these are independent event.<\/li>\r\n \t<li style=\"text-align: justify\">Tossing a coin is a random experiment,<span style=\"text-align: initial;font-size: 1em\"> if one <\/span>get<span style=\"text-align: initial;font-size: 1em\"> head on first toss as well as second toss. One can see that both tosses of a coin are not affecting the outcome of each other, so these are <\/span>independent<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Drawing a card from a pack of card<span style=\"text-align: initial;font-size: 1em\"> with replacement is <\/span>independent<span style=\"text-align: initial;font-size: 1em\"> event. Every time <\/span>total<span style=\"text-align: initial;font-size: 1em\"> number of cards remain <\/span>same<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 4.8 Equally likely events<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">All possible event of a random experiment have same numerical value of probability of occurring events. These events are known as equally likely events with equal probability of occurrence.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cEqually likely events shows equal probability for all events. There is no preference for any event, all have equal preference\u201d.<\/p>\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Tossing a coin is a random experiment. Possible outcomes are head and tail. Possible event are getting head and getting tail. One has total possible outcome as 2.<\/p>\r\n&nbsp;\r\n\r\nProbability of getting head is \u00bd.\r\n\r\n&nbsp;\r\n\r\nProbability of getting tail is \u00bd.\r\n\r\n&nbsp;\r\n\r\nBoth have equal probability and equal preference. So, events are equally likely events.\r\n\r\n&nbsp;\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Throwing a die is a random experiment with six possible outcomes and events. All possible event have equal probability i.e. 1\/6. So all events have equal preference to happening, known as equally likely events.<\/p>\r\n&nbsp;\r\n\r\n<strong>4.9 Impossible event<\/strong>\r\n\r\n&nbsp;\r\n\r\nIf an event under a random experiment cannot occur, that event is known as an impossible event.\r\n\r\n&nbsp;\r\n\r\n\u201cNumerical measure of an event is zero then the event is impossible event\u201d.\r\n\r\n&nbsp;\r\n\r\nFor an impossible event E,\r\n<p style=\"text-align: center\">P(E) = 0.<\/p>\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If tossing a dice is a random experiment. E is an event of outcome greater than 6.We know that dice have six faces only so there is no chance of happening this event. Hence event E is an impossible event.<\/p>\r\n&nbsp;\r\n\r\n<strong>4.10 Certain event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If there is an absolute surety on the chance of occurrence of an event, event is known as certain event. Suppose if it is Thursday, the probability that tomorrow is Friday is certain event with one probability.<\/p>\r\n&nbsp;\r\n\r\n\u201cNumerical measure of an event is one, then event is certain event\u201d.\r\n\r\n&nbsp;\r\n\r\nFor a sure and certain event E:\r\n<p style=\"text-align: center\">P(E) = 1.<\/p>\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hitting a dart is a random experiment and throwing a dart at <\/span>center<span style=\"text-align: initial;font-size: 1em\"> is an event on the dartboard. Imagine that this is only one thing in the universe and dart hit at the center, there is only one chance to throw then this event is <\/span>sure<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 4.11 Complementary event<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Non-happening of an event (A) is known as complementary event and it is denoted by . Sum of happening and non-happening event will be one.<\/p>\r\n&nbsp;\r\n\r\n?(?) + ?(?? ) = 1.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Tossing a coin is a random experiment and we are interested in getting head. If head appears then it is a happening event, getting tail is non-happening event and complementary event.<\/li>\r\n \t<li style=\"text-align: justify\">Rolling a dice<span style=\"text-align: initial;font-size: 1em\"> is a random experiment and getting 6 on dice is <\/span>event<span style=\"text-align: initial;font-size: 1em\">. Getting value other <\/span>then<span style=\"text-align: initial;font-size: 1em\"> 6 or not getting 6 is <\/span>non-happening<span style=\"text-align: initial;font-size: 1em\"> event and <\/span>complementary<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 5.\u00a0 <\/strong><strong>Probability Approach<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">After understanding the statement of the problem, next move is to calculate the numerical value of uncertainty. To calculate the numerical value of the probability, several types of approach and methods available in the theory of probability. Some of them are discussed here. Most of the time, three types of approach are used to solve probability problem in numerical form. In this section, these three approach are defined. Basically, three approach of probability are also used as a definition of the probability.<\/p>\r\n&nbsp;\r\n\r\n<strong>a.<\/strong>\u00a0\u00a0 <strong>Mathematical approach<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This definition of probability given by James Bernoulli who was the first person of obtaining numerical measure of uncertainty. This definition of probability is also known as classical or priori probability.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">If \u201cF\u201d denotes the favorable outcome of an event (E) of a random experiment and \u201cT\u201d denotes the total possible outcome. The probability of happening an event (E) is given by:<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-171\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80.png\" alt=\"\" width=\"399\" height=\"56\" \/>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0Note: <\/strong><span style=\"text-align: initial;font-size: 1em\">All possible outcomes in <\/span>mathematical<span style=\"text-align: initial;font-size: 1em\"> approach must be exhaustive, mutually exclusive and equally likely.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Example<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Suppose a random experiment of randomly selecting a card from a deck of cards of 52 has equally likely outcomes.<\/span><\/p>\r\n\r\n<div>\r\n\r\n&nbsp;\r\n\r\nLet event A = {king}, probability of an event A = 4\/52\r\n\r\n&nbsp;\r\n\r\nB = {jack, queen, ace}, probability of an event B = 12\/52.\r\n\r\n&nbsp;\r\n\r\nFavorable outcomes are 4, 12 on both event.\r\n\r\n&nbsp;\r\n\r\nb.\u00a0\u00a0<strong>Statistical approach<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Probability of happening an event E under a random experiment perform repeatedly under regular and identical circumstances is the ratio of the number of times event occurred to the number of\u00a0 trial. This\u00a0limit must be unique and finite.<\/p>\r\n&nbsp;\r\n\r\nProbability of happening event E given by:\r\n\r\n<img class=\"aligncenter size-full wp-image-172\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81.png\" alt=\"\" width=\"335\" height=\"57\" \/>\r\n\r\nThis approach of probability is also known as <strong>Empirical<\/strong> approach of probability due it\u2019s repetitive behavior.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">A survey conducted in Bathinda district of Punjab state to observe that how many people wearing helmets or not. Total population of Bathinda district is approx. 285813. Let the 40000 use regularly helmet. Probability of use of helmet on regular basis is 40000\/285813= 0.139951.<\/p>\r\n&nbsp;\r\n\r\n<strong>c.<\/strong>\u00a0<strong>Subjective approach:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Subjective approach is completely biased. Numerical measure of this approach cannot be possible because it is based on biasedness, based on person\u2019s knowledge, beliefs, experience and intuition.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose one old person assumes the chance of cold in winter will be normal and not too cold. One business man observes the chance of no profit in the business this year.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Above example of subjective approach will vary person to person. This approach is not quite enough to measure probability numerically.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 6. Axioms of Probability<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Numerical measure of uncertainty to respective event is called probability. For accurate and best result of an event as much as possible probability must follow some axioms.<\/p>\r\n&nbsp;\r\n\r\nAxioms of probability are given as:\r\n\r\n&nbsp;\r\n\r\n<strong>a)\u00a0 Axioms of non-negativity:<\/strong>\r\n\r\n&nbsp;\r\n\r\n\u201cProbability of an event must be non-negative\u201d.\r\n\r\n&nbsp;\r\n\r\nSuppose E is any event of a random experiment, probability of E is:\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">?(?) \u2265 0<\/p>\r\n&nbsp;\r\n\r\nProbability lies between one and zero, probability must be non-negative but less than one.\r\n\r\n&nbsp;\r\n\r\n0 \u2264 ?(?) \u2264 1.\r\n\r\n&nbsp;\r\n\r\n<strong>b)\u00a0 Axioms of certainty:<\/strong>\r\n\r\n&nbsp;\r\n\r\n\u201cProbability sum of all possible outcomes of a random experiment must be one\u201d.\r\n\r\n&nbsp;\r\n\r\nSuppose E is an event of a random experiment, probability sum of all possible event is:\r\n\r\n&nbsp;\r\n\r\n?(?) = 1.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If probability of getting head at first time of tossing a coin is 0.5 and probability of getting tail on first time of tossing a coin is 0.5. Total of getting head or tail is 1.<\/p>\r\n&nbsp;\r\n\r\n<strong>c)<\/strong>\u00a0\u00a0\u00a0 <strong>Axioms of additivity:<\/strong>\r\n\r\n&nbsp;\r\n\r\n\u201cProbability of union of all events is equal to the sum of the probability of all distinct event\u201d.\r\n\r\n<img class=\"size-full wp-image-173 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82.png\" alt=\"\" width=\"585\" height=\"112\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">7. Permutations<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If a probability problem is <\/span>related<span style=\"text-align: initial;font-size: 1em\"> some specific type arrangement like <\/span>arrangement<span style=\"text-align: initial;font-size: 1em\"> of letter of a word, arrangement of persons in a queue etc. Permutations are used to solve <\/span>these type<span style=\"text-align: initial;font-size: 1em\"> of arrangement problem.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0\u00a0Mathematical form of permutation<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Special arrangements of items defined by permutations. When selecting m items from the N items, possible permutations are given by this formula:<\/p>\r\n<img class=\"size-full wp-image-174 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83.png\" alt=\"\" width=\"775\" height=\"215\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>8. Combination<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If a probability problem is related to selection of some unit from a huge amount of same unit like selection a ball from a bag, selecting a card from a deck of card etc. Combinations are used to solve these type of selection problem.<\/p>\r\n&nbsp;\r\n\r\n<strong>Mathematical form of combination<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Selection of items from group of items defined by combination. When selecting m items from the N items, possible combination is given by:<\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-175\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-84.png\" alt=\"\" width=\"166\" height=\"61\" \/>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Example<\/strong>\r\n\r\n<img class=\"size-full wp-image-176 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85.png\" alt=\"\" width=\"607\" height=\"45\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div>\r\n\r\n&nbsp;\r\n\r\nPossible combinations of selecting 2 marble from a bag of 6 marble = 2!4!6! = 15.\r\n\r\n&nbsp;\r\n\r\n<strong>Question 1<\/strong>\r\n\r\n&nbsp;\r\n\r\nOne unbiased dice is thrown. Find the probability:\r\n\r\n&nbsp;\r\n\r\na.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows 5\r\n\r\nb.\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows even number\r\n\r\nc.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows odd number\r\n\r\n<img class=\"aligncenter size-full wp-image-177\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-86.png\" alt=\"\" width=\"194\" height=\"101\" \/>\r\n\r\n<strong>Answer<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">One unbiased dice is thrown randomly. Possible outcomes are 1, 2, 3, 4, 5 and 6. Mathematical approach is used to find probability for given event a, b, and c.<\/p>\r\n&nbsp;\r\n\r\nFor an event a:\r\n\r\n&nbsp;\r\n\r\nAs\u00a0 dice\u00a0 shows\u00a0 five\u00a0 only\u00a0 one\u00a0 time.\u00a0 Favorable\u00a0 outcome\u00a0 is\u00a0 1\u00a0 and\u00a0 total\u00a0 possible\u00a0 outcomes\u00a0 are\u00a0 6.\r\n\r\nProbability of event a is given by:\r\n\r\n<img class=\"aligncenter size-full wp-image-178\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-87.png\" alt=\"\" width=\"189\" height=\"54\" \/>\r\n\r\n&nbsp;\r\n\r\nFor event b:\r\n\r\n&nbsp;\r\n\r\nAs dice has three even number i.e. 2, 4, 6. Favorable outcomes are 3 and total outcomes are 6.\r\n\r\nProbability of event b is given by:\r\n\r\n<img class=\"aligncenter size-full wp-image-180\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-88.png\" alt=\"\" width=\"190\" height=\"59\" \/>\r\n\r\nFor event c:\r\n\r\n&nbsp;\r\n\r\nAlso dice has three odd number i.e. 1, 3, 5. Favorable outcomes are 3 and total outcomes are 6.\r\n\r\nProbability of event c is given by:\r\n\r\n<img class=\"aligncenter size-full wp-image-181\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-89.png\" alt=\"\" width=\"179\" height=\"54\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Question 2<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">Suppose a coin is tossed two times.\u00a0 Find the probability of getting head one first time?<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Answer<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-182\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-90.png\" alt=\"\" width=\"177\" height=\"134\" \/>\r\n\r\nCoin has two possible outcomes one is head and second is tail.\r\n\r\n&nbsp;\r\n\r\nAfter tossing the coin two times. There are four possible outcomes\r\n\r\n&nbsp;\r\n\r\ni.e. HH, HT, TH, TT.\r\n\r\n&nbsp;\r\n\r\nFavorable outcomes for event are 2 (HH, HT) and total outcomes are 4.\r\n\r\n&nbsp;\r\n\r\nProbability of getting head on first place or on first coin is\r\n\r\n<img class=\"aligncenter size-full wp-image-183\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-91.png\" alt=\"\" width=\"152\" height=\"49\" \/>\r\n\r\n<strong>Question 3<\/strong>\r\n\r\n&nbsp;\r\n\r\nSuppose a bag has 20 with different colors.\r\n\r\n&nbsp;\r\n\r\nThere are 5 green marbles, 6 blue marbles and 9 red marbles.\r\n\r\n&nbsp;\r\n\r\nLet us consider that two marbles are selected randomly.\r\n\r\n&nbsp;\r\n\r\nFind the probability of getting two red marbles, getting one red\r\n\r\n&nbsp;\r\n\r\nand one blue marbles, getting one blue and one green ball.\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-184\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-92.png\" alt=\"\" width=\"147\" height=\"178\" \/>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Answer<\/strong>\r\n\r\n&nbsp;\r\n\r\nProbability of getting two red marbles:\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Bag contains 9 red marbles and total marbles are 20. We are interested in to find out the probability that two marbles are selected from 20 marbles and it must be red so these two selected marbles will be from 9 red marbles. Combination will be used to find out probability as:<\/p>\r\n<img class=\"aligncenter size-full wp-image-185\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-93.png\" alt=\"\" width=\"120\" height=\"151\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-186\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-94.png\" alt=\"\" width=\"212\" height=\"55\" \/>\r\n\r\n&nbsp;\r\n\r\nAs bag contains 9 red marbles and 6 blue marbles to find out the probability of one red marble and one blue marble. The formula is\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-187\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95.png\" alt=\"\" width=\"765\" height=\"502\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>9. Summary<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This module will help learner to understand basic of probability and terminology that is essential for better understanding of the problem. Probability approach is discusses that helps to transform the possibility into numerical results. In this module, main concern is only to introduce you keyconcepts and simple probability approach and application of probability. Advance probability problems will be discussed in the further probability module.<\/p>\r\n\r\n<\/div>\r\n<ol start=\"10\">\r\n \t<li><strong> Suggested Readings<\/strong><\/li>\r\n<\/ol>\r\nAgresti, A. and B. Finlay, <em>Statistical Methods for the Social Science, 3<\/em><em>rd<\/em> <em>Edition, Prentice Hall<\/em>, 1997.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Daniel, W. W. and C. L. Cross, C. L., <em>Biostatistics<\/em>: <em>A Foundation for Analysis in the Health Sciences,<\/em> 10th Edition<em>,<\/em> John Wiley &amp; Sons, 2013.<\/p>\r\n&nbsp;\r\n\r\nHogg, R. V., J. Mckean and A. Craig, <em>Introduction to Mathematical Statistics<\/em>, Macmillan Pub. Co. Inc., 1978.\r\n\r\n&nbsp;\r\n\r\nMeyer, P. L., <em>Introductory Probability and Statistical Applications<\/em>, Oxford &amp; IBH Pub, 1975.\r\n\r\n&nbsp;\r\n\r\nStephens, L. J., <em>Schaum\u2019s Series Outline: Beginning Statistics,<\/em> 2nd Edition<em>,<\/em> McGraw Hill, 2006.\r\n\r\n&nbsp;\r\n\r\nTriola, M. F., <em>Elementary Statistics<\/em>, 13th\u00a0 Edition, Pearson, 2017.\r\n\r\n&nbsp;\r\n\r\nWeiss, N. A., <em>Introductory Statistics<\/em>, 10th Edition, Pearson, 2017.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Introduction to Probability<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/u5sFDxoIvvo\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nOne can refer to the following links for further understanding of the statistics terms.\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf\">http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html\">http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf\">http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/stats.oecd.org\/glossary\/\">https:\/\/stats.oecd.org\/glossary\/<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary\">http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm\">https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A\">https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A<\/a>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/u5sFDxoIvvo\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p>\u00a0 \u00a0\u00a0<strong>Learning Objectives<\/strong><\/p>\n<ul>\n<li><strong>Introduction<\/strong><\/li>\n<li><strong>Terminology<\/strong><\/li>\n<li><strong>Types of Events<\/strong><\/li>\n<li><strong>Probability Approach<\/strong><\/li>\n<li><strong>Axioms of Probability<\/strong><\/li>\n<li><strong>Permutations<\/strong><\/li>\n<li><strong>Combination<\/strong><\/li>\n<li><strong>Summary<\/strong><\/li>\n<li><strong>Suggested Readings<\/strong><\/li>\n<\/ul>\n<\/div>\n<p><span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0\u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">The objective of this module is to introduce key concepts that help to understand <\/span>probability<span style=\"text-align: justify;font-size: 1em\"> problem in simple words and different approaches to find <\/span>numerical<span style=\"text-align: justify;font-size: 1em\"> measure of probability for different probability problem. Examples and some images are used for <\/span>better<span style=\"text-align: justify;font-size: 1em\"> explanation of this topic in an easy manner.<\/span><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>2.<\/strong>\u00a0\u00a0 <strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Probability measures the uncertainty of an event in an experiment. Theory of probability is an important branch of mathematics and statistics that provides numerical measure of the probability. Probability is also used in our daily life. Likely, possible, high chance of use of like words in day-to-day life to express uncertainty of happening an event. For example: possibly, it will rain today and there is a high chance to get my dream job in next two months.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">According to Ya-Lin-Chou \u201cProbability is the science of decision making with calculated risks in the face of uncertainty\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Probability phenomenon is mostly observed in business, economics, social science, daily life and actuaries. Probability has many application in economics, business and actuaries because there are terminologies such as risk, taxes, profit, loss etc. These terminologies express uncertainty of an event under an experiment. Suppose an investor invest money in money market. There are chances that either investor will get more money as profit of investment or will face loss in the investment. This shows uncertainty, probability terminology can be used here.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A real life exercise of number plate of Victorian is shown in Figure 1, you will be able to solve these types of exercises after completing this module on probability.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-167\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78.png\" alt=\"\" width=\"430\" height=\"302\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78.png 430w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78-300x211.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78-65x46.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78-225x158.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-78-350x246.png 350w\" sizes=\"auto, (max-width: 430px) 100vw, 430px\" \/><\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">3<\/strong><span style=\"text-align: initial;font-size: 1em\">.<\/span><strong style=\"text-align: initial;font-size: 1em\"> Terminology<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There is a need to be aware <\/span>about<span style=\"text-align: initial;font-size: 1em\"> some basic concepts of probability before going to <\/span>numerical<span style=\"text-align: initial;font-size: 1em\"> measure of probability. In this section, <\/span>focus<span style=\"text-align: initial;font-size: 1em\"> will be on key concepts to understand a probability problem in words and convert into numerical form to solve it by using <\/span>different<span style=\"text-align: initial;font-size: 1em\"> approach of probability according to the features of the problems.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>Key concepts are given as:<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.1 Random Experiment<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Before conducting an experiment there is no possibility to get outcome of any problem. Experiment is a process to observe an uncertainty in the problem.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cRandom experiment is an experiment in which outcome of an trial or event under some identical circumstances are not unique but it will be among all the possible outcomes.\u201d<\/p>\n<p>&nbsp;<\/p>\n<p>List of some examples of random experiment is given below:<\/p>\n<ul>\n<li>Tossing a coin.<\/li>\n<li>Rolling a dice.<\/li>\n<li>Observe risk<span style=\"text-align: initial;font-size: 1em\"> of an investor.<\/span><\/li>\n<li>Selecting a card from a deck of card.<\/li>\n<li>Selecting a ball from a group of balls.<\/li>\n<li>Arrangement<span style=\"text-align: initial;font-size: 1em\"> of persons in a queue.<\/span><\/li>\n<li>Observe the number of Xiaomi android phones sold by Amazon in 2017.<\/li>\n<li>Selecting a student from a classroom.<\/li>\n<li>Observe the number of wickets in a cricket match.<\/li>\n<li>Observe product<span style=\"text-align: initial;font-size: 1em\"> of a process in a manufacturing company. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Observe a bulb is either fuse or not.<\/span><\/li>\n<li>Observe a person who speaks either truth or lie.<\/li>\n<li>Observe the number of car<span style=\"text-align: initial;font-size: 1em\"> accident in <\/span>particular<span style=\"text-align: initial;font-size: 1em\"> place. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Observe the number of birth in India in <\/span>year<span style=\"text-align: initial;font-size: 1em\"> 2017.<\/span><\/li>\n<li>Observe the share market prices.<\/li>\n<\/ul>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 3.2 Outcome<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">After conduct of a random experiment, now to observe the uncertainty of a problem, there will be some results or outcome of that experiment. Outcome gives us an idea about all possible results of that experiment.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cOutcome is a result of a random experiment\u201d.<\/p>\n<p>List of some outcomes of random experiment is given as:<\/p>\n<ul>\n<li>Outcome of tossing a coin:\u00a0 head (H) or tail (T).<\/li>\n<li>Outcome of rolling a dice: 1\/ 2\/ 3\/ 4\/ 5\/ 6.<\/li>\n<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of selecting a ball from a group of different colors balls like blue, red, white: blue\/ red\/ white.<\/span><\/li>\n<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of <\/span>risk<span style=\"text-align: initial;font-size: 1em\"> of an investor: high risk\/ neutral risk\/ low risk\/ no risk.<\/span><\/li>\n<li style=\"text-align: justify\">Outcome of selecting a card from a deck of card: 1\/ 2\/ 3\/\u2026. \/10\/ J\/ Q\/ K\/ A\/ club\/ spade\/ diamond\/ heart\/ red\/ black.<\/li>\n<li style=\"text-align: justify\">Outcome<span style=\"text-align: initial;font-size: 1em\"> of <\/span>working<span style=\"text-align: initial;font-size: 1em\"> condition of a bulb: Fuse\/not fuse. <\/span>Outcome<span style=\"text-align: initial;font-size: 1em\"> of a person speaking tendency; truth\/ lie.<\/span><\/li>\n<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of observing car accident in a state: 1, 2, \u2026\u2026<\/span><\/li>\n<li>Outcome<span style=\"text-align: initial;font-size: 1em\"> of product quality in a manufacturing company: defective\/ non-defective.<\/span><\/li>\n<\/ul>\n<\/div>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a03.3 Sample space:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">After identifying the result of the experiment, there is a need to arrange all possible outcome. In basic mathematics, the definition of <\/span><strong style=\"text-align: initial;font-size: 1em\">set<\/strong><span style=\"text-align: initial;font-size: 1em\"> was described. Set is a group of well-defined objects <\/span>that related<span style=\"text-align: initial;font-size: 1em\"> to each other under <\/span>same<span style=\"text-align: initial;font-size: 1em\"> consideration. Here, <\/span>result<span style=\"text-align: initial;font-size: 1em\"> of an experiment is well defined different result under <\/span>same<span style=\"text-align: initial;font-size: 1em\"> consideration, so set is used as sample space to arrange all possible outcomes.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cSample space is set of all possible outcomes of an experiment\u201d.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">List of some examples of sample space <\/span>are<span style=\"text-align: initial;font-size: 1em\">:<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Sample space for tossing a coin: {H, T}.<\/span><\/li>\n<li style=\"text-align: justify\">Sample space for rolling a dice: {1, 2, 3, 4, 5, 6}.<\/li>\n<li style=\"text-align: justify\">Sample space for selecting a card from a deck of card: {1, 2, \u2026, 10, A, K, Q, J}.<\/li>\n<li style=\"text-align: justify\">Sample space for selecting a ball from a group of balls of different<span style=\"text-align: initial;font-size: 1em\"> color like blue, green and red: {Blue ball, red ball, green ball}.<\/span><\/li>\n<li style=\"text-align: justify\">Sample space of observe risk<span style=\"text-align: initial;font-size: 1em\"> of an investor: {high, low, neutral, no}.<\/span><\/li>\n<li style=\"text-align: justify\">Sample space of product that produces by a manufacturing company: {defective, non<span style=\"text-align: initial;font-size: 1em\">-defective}.<\/span><\/li>\n<li style=\"text-align: justify\">Sample space of observe a bulb: {fuse, non-fuse}.<\/li>\n<li style=\"text-align: justify\">Sample space of observing person talk: {truth, lie}.<\/li>\n<li style=\"text-align: justify\">Sample space for observe car accident in a state: {1, 2, 3\u2026}.<\/li>\n<li style=\"text-align: justify\">Sample space for observe<span style=\"text-align: initial;font-size: 1em\"> the number of wickets in a cricket match: {1, 2, \u2026, 22}.<\/span><\/li>\n<\/ul>\n<div>\n<p><strong>\u00a0 \u00a0 3.4 Event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If, one is interested in particular results among all possible results of a random experiment, these particular results can be obtained in terms of events.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cEvent is a particular performance of a random experiment\u201d.<\/p>\n<p>List of some examples of event:<\/p>\n<ul>\n<li>Suppose a random experiment is tossing a coin two times.<\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 Sample space<\/strong>: {HH, HT, TH, TT}<\/p>\n<p><strong>Possible event<\/strong>: getting head both times, getting tail both times, getting one head and tail of a coin.<\/p>\n<ul>\n<li>Suppose rolling a die two times is a random Experiment.<\/li>\n<\/ul>\n<\/div>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Sample space<\/strong><span style=\"text-align: initial;font-size: 1em\">: {(1,1), (1,2),\u2026\u2026\u2026,(1,6), (2,1), (2,2), \u2026\u2026..,(2,6), (3,1), (3,2),\u2026\u2026.,(3,6), (4,1),\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">(4,2), \u2026\u2026..,(4,6), (5,1), (5,2),\u2026\u2026.,(5,6), (6,1), (6,2),\u2026..,(6,6)}.<\/span><\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Possible event<\/strong><span style=\"text-align: initial;font-size: 1em\">: Getting <\/span>same<span style=\"text-align: initial;font-size: 1em\"> number both <\/span>times ,<span style=\"text-align: initial;font-size: 1em\"> getting <\/span>odd<span style=\"text-align: initial;font-size: 1em\"> number at <\/span>first<span style=\"text-align: initial;font-size: 1em\"> time and even number at <\/span>second<span style=\"text-align: initial;font-size: 1em\"> time, getting <\/span>odd<span style=\"text-align: initial;font-size: 1em\"> number at first time, getting <\/span>sum<span style=\"text-align: initial;font-size: 1em\"> of both is <\/span>even ,<span style=\"text-align: initial;font-size: 1em\"> getting <\/span>sum<span style=\"text-align: initial;font-size: 1em\"> of both <\/span>time<span style=\"text-align: initial;font-size: 1em\"> is <\/span>odd ,<span style=\"text-align: initial;font-size: 1em\"> many more event may occur.<\/span><\/p>\n<div>\n<ul>\n<li>Selecting a card from a deck of cards is an random experiment.<\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 Sample space<\/strong>: {1, 2, \u2026\u2026,10, A, K, Q, J}.<\/p>\n<p style=\"text-align: justify\"><strong>Possible event<\/strong>: Getting a spade, getting a club, getting a heart , getting a diamond, getting a face card, getting a king, getting a queen , many more event may occur respect to sample space of random experiment.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4<\/strong><strong>.<\/strong><strong> Types of Events<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">There are many types of event according to the happening of event and outcome. In this section, types of event are defined.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.1 Simple event:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose a particular event happens among all possible events. Result of particular event is at most one, then event is called simple event.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cAn event that shows exactly one outcome of a random experiment is a simple event\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose a quality analyst analysis two processes\u2019 product in a random experiment and classify the process product as defective product and non-defective product.<\/p>\n<p>&nbsp;<\/p>\n<p>Sample space: {nn, nd, dn, dd}<\/p>\n<p>&nbsp;<\/p>\n<p>where d is for defective product and n is for non-defective product.<\/p>\n<p>&nbsp;<\/p>\n<p>Let E is an event.<\/p>\n<p>&nbsp;<\/p>\n<p>Event E : neither product is defective.<\/p>\n<p>&nbsp;<\/p>\n<p>E ={nn}; as E has only one outcome, E is a simple event.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Example<\/strong><\/p>\n<\/div>\n<div>\n<p>\u00a0 \u00a0 Tossing a coin two times in a random experiment.<\/p>\n<p>&nbsp;<\/p>\n<p>Sample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail<\/p>\n<p>&nbsp;<\/p>\n<p>Let E be an event: getting head both time.<\/p>\n<p>&nbsp;<\/p>\n<p>E = {HH}; E has only one outcome, E is a simple event.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.2 Compound event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose a particular event happens among all possible events. Result of particular event is at least one outcome that event is called a compound event.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cA event shows more than one outcome of a random experiment is a compound event\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose a quality analyst analysis two processes\u2019 product quality in an random experiment and classify the process product as defective product and non-defective product.<\/p>\n<p>&nbsp;<\/p>\n<p>Sample space : {nn, nd, dn, dd}<\/p>\n<p>&nbsp;<\/p>\n<p>where: d is for defective product and n is for non-defective product.<\/p>\n<p>&nbsp;<\/p>\n<p>Let E is an event.<\/p>\n<p>&nbsp;<\/p>\n<p>Event E : at least one product is defective.<\/p>\n<p>&nbsp;<\/p>\n<p>E: {nd, dn, dd}; E has more than one outcome. So, E is a compound event.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Tossing a coin two times is a random experiment.<\/p>\n<p>&nbsp;<\/p>\n<p>Sample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail<\/p>\n<p>&nbsp;<\/p>\n<p>Let E be an event getting head at least one time.<\/p>\n<p>&nbsp;<\/p>\n<p>E: {HH, HT, TH}; E has more than one outcome. So, E is a compound event.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.3 Exhaustive event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">All possible results of an experiment <\/span>is<span style=\"text-align: initial;font-size: 1em\"> termed as an exhaustive event.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u201cExhaustive event is the total outcome of a random experiment\u201d.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">List of some examples are:<\/span><\/p>\n<ul>\n<li><span style=\"text-align: initial;font-size: 1em\">Tossing of one coin is a random experiment, exhaustive cases are 2. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Throwing a dice, exhaustive cases are 6.<\/span><\/li>\n<li>Throwing two dices, exhaustive cases are 62.<\/li>\n<li>Tossing two coins, exhaustive cases are 22.<\/li>\n<li>Selecting a card, exhaustive cases are 52.<\/li>\n<\/ul>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 4.4 Dependent event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Two particular events E and F occurred in a trail. Result of one event are affecting the result of second event. Event may be either E or F, these events will be dependent event.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cEvent is said to be dependent event if happening of one event in one trail is affected by happening of similar event in next trail\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Examples:<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Drawing a card from a pack of card followed by one after another draw is a random experiment. Let us consider E is an event of drawing black card. If first drawn card is not replaced in the deck of card while second card is drawn, then the second time drawn card will be dependent on the first drawn card. After this total number of cards are decreasing followed by one after another draw. So, E is a dependent event.<\/li>\n<li style=\"text-align: justify\">Let us consider a bag with five marbles, four are red and one is blue. If a marble is selected at random from the bag and not replaced again into the bag, the chance of occurrence of the blue ball will keep increasing with each consecutive draw.<\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 4.5 Favorable event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Outcomes of a particular event are favorable to that events. These outcomes of events are considered as favorable event.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cFavorable event is the number of outcome favor to an event\u201d.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Examples<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Tossing\u00a0 two\u00a0 coins\u00a0 is\u00a0 a<span style=\"text-align: initial;font-size: 1em\">\u00a0 <\/span>random\u00a0 experiment\u00a0 and\u00a0 getting\u00a0 head\u00a0 on\u00a0 both\u00a0 coins\u00a0 is\u00a0 an\u00a0 event<span style=\"text-align: initial;font-size: 1em\">.\u00a0<\/span>Exhaustive<span style=\"text-align: initial;font-size: 1em\"> event is 22 = 4 and <\/span>favorable<span style=\"text-align: initial;font-size: 1em\"> event is 1 (i.e. head and head).<\/span><\/li>\n<li style=\"text-align: justify\">Throwing two dice is an random<span style=\"text-align: initial;font-size: 1em\"> experiment and getting <\/span>same<span style=\"text-align: initial;font-size: 1em\"> number on both <\/span>side<span style=\"text-align: initial;font-size: 1em\"> is an event, <\/span>favorable<span style=\"text-align: initial;font-size: 1em\"> event is 6 {(1,1), (2,2), (3,3), (4,4), ( 5,5), (6,6)}.<\/span><\/li>\n<\/ul>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a04.6 Mutually exclusive event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u201cEvents can\u2019t occur simultaneously in <\/span>same<span style=\"text-align: initial;font-size: 1em\"> trial are mutually exclusive events\u201d.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Suppose A and B are two events. If A and B are <\/span>mutually<span style=\"text-align: initial;font-size: 1em\"> exclusive event, their <\/span>venn<span style=\"text-align: initial;font-size: 1em\"> diagram is given\u00a0in Figure 2.<\/span><span style=\"text-align: initial;font-size: 1em\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-168\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-79.png\" alt=\"\" width=\"237\" height=\"145\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-79.png 237w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-79-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-79-225x138.png 225w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/span><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\">Figure 2<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Head and tail are two outcomes of random experiment of a coin. Suppose A and B are two events of getting head and getting tail respectively. Both event cannot occur in a single trail, so A and B are mutually exclusive event.<\/li>\n<li style=\"text-align: justify\">1, 2, 3, 4, 5, 6 are the possible six outcome<span style=\"font-size: 1em\"> of <\/span>random<span style=\"font-size: 1em\"> experiment in throwing <\/span>a dice<span style=\"font-size: 1em\">. All of these outcomes cannot occur in a single trail, so they are all mutually exclusive event.<\/span><\/li>\n<li style=\"text-align: justify\">A switch cannot be on\/ off at same<span style=\"text-align: initial;font-size: 1em\"> time. Switching a switch button is a random experiment that has two possible outcomes on and off that switch is on while the second one shows switch is off. Both <\/span>event<span style=\"text-align: initial;font-size: 1em\"> cannot occur at <\/span>same<span style=\"text-align: initial;font-size: 1em\"> time, so A and B are <\/span>mutually<span style=\"text-align: initial;font-size: 1em\"> exclusive event.<\/span><\/li>\n<\/ul>\n<\/div>\n<p><strong>\u00a0 \u00a0 4.7 Independent event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Two particular event E and F are occurred in a trial. Result of one event is not affecting the result of second event. Event may be either E or F, so these events will be independent event.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cEvent is said to be independent event if happening of one event in one trail is not affected by happening of similar event in next trial\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Throwing a dice is a random experiment, if one get six in first throw as well as in second throw. One can see that both throws of a dice are not affecting the outcome of each other, so these are independent event.<\/li>\n<li style=\"text-align: justify\">Tossing a coin is a random experiment,<span style=\"text-align: initial;font-size: 1em\"> if one <\/span>get<span style=\"text-align: initial;font-size: 1em\"> head on first toss as well as second toss. One can see that both tosses of a coin are not affecting the outcome of each other, so these are <\/span>independent<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/li>\n<li style=\"text-align: justify\">Drawing a card from a pack of card<span style=\"text-align: initial;font-size: 1em\"> with replacement is <\/span>independent<span style=\"text-align: initial;font-size: 1em\"> event. Every time <\/span>total<span style=\"text-align: initial;font-size: 1em\"> number of cards remain <\/span>same<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 4.8 Equally likely events<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">All possible event of a random experiment have same numerical value of probability of occurring events. These events are known as equally likely events with equal probability of occurrence.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cEqually likely events shows equal probability for all events. There is no preference for any event, all have equal preference\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Tossing a coin is a random experiment. Possible outcomes are head and tail. Possible event are getting head and getting tail. One has total possible outcome as 2.<\/p>\n<p>&nbsp;<\/p>\n<p>Probability of getting head is \u00bd.<\/p>\n<p>&nbsp;<\/p>\n<p>Probability of getting tail is \u00bd.<\/p>\n<p>&nbsp;<\/p>\n<p>Both have equal probability and equal preference. So, events are equally likely events.<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Throwing a die is a random experiment with six possible outcomes and events. All possible event have equal probability i.e. 1\/6. So all events have equal preference to happening, known as equally likely events.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.9 Impossible event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>If an event under a random experiment cannot occur, that event is known as an impossible event.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cNumerical measure of an event is zero then the event is impossible event\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p>For an impossible event E,<\/p>\n<p style=\"text-align: center\">P(E) = 0.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If tossing a dice is a random experiment. E is an event of outcome greater than 6.We know that dice have six faces only so there is no chance of happening this event. Hence event E is an impossible event.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.10 Certain event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If there is an absolute surety on the chance of occurrence of an event, event is known as certain event. Suppose if it is Thursday, the probability that tomorrow is Friday is certain event with one probability.<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cNumerical measure of an event is one, then event is certain event\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p>For a sure and certain event E:<\/p>\n<p style=\"text-align: center\">P(E) = 1.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hitting a dart is a random experiment and throwing a dart at <\/span>center<span style=\"text-align: initial;font-size: 1em\"> is an event on the dartboard. Imagine that this is only one thing in the universe and dart hit at the center, there is only one chance to throw then this event is <\/span>sure<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 4.11 Complementary event<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Non-happening of an event (A) is known as complementary event and it is denoted by . Sum of happening and non-happening event will be one.<\/p>\n<p>&nbsp;<\/p>\n<p>?(?) + ?(?? ) = 1.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Tossing a coin is a random experiment and we are interested in getting head. If head appears then it is a happening event, getting tail is non-happening event and complementary event.<\/li>\n<li style=\"text-align: justify\">Rolling a dice<span style=\"text-align: initial;font-size: 1em\"> is a random experiment and getting 6 on dice is <\/span>event<span style=\"text-align: initial;font-size: 1em\">. Getting value other <\/span>then<span style=\"text-align: initial;font-size: 1em\"> 6 or not getting 6 is <\/span>non-happening<span style=\"text-align: initial;font-size: 1em\"> event and <\/span>complementary<span style=\"text-align: initial;font-size: 1em\"> event.<\/span><\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 5.\u00a0 <\/strong><strong>Probability Approach<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">After understanding the statement of the problem, next move is to calculate the numerical value of uncertainty. To calculate the numerical value of the probability, several types of approach and methods available in the theory of probability. Some of them are discussed here. Most of the time, three types of approach are used to solve probability problem in numerical form. In this section, these three approach are defined. Basically, three approach of probability are also used as a definition of the probability.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>a.<\/strong>\u00a0\u00a0 <strong>Mathematical approach<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This definition of probability given by James Bernoulli who was the first person of obtaining numerical measure of uncertainty. This definition of probability is also known as classical or priori probability.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If \u201cF\u201d denotes the favorable outcome of an event (E) of a random experiment and \u201cT\u201d denotes the total possible outcome. The probability of happening an event (E) is given by:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-171\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80.png\" alt=\"\" width=\"399\" height=\"56\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80.png 399w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80-300x42.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80-225x32.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-80-350x49.png 350w\" sizes=\"auto, (max-width: 399px) 100vw, 399px\" \/><\/p>\n<\/div>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0Note: <\/strong><span style=\"text-align: initial;font-size: 1em\">All possible outcomes in <\/span>mathematical<span style=\"text-align: initial;font-size: 1em\"> approach must be exhaustive, mutually exclusive and equally likely.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Suppose a random experiment of randomly selecting a card from a deck of cards of 52 has equally likely outcomes.<\/span><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>Let event A = {king}, probability of an event A = 4\/52<\/p>\n<p>&nbsp;<\/p>\n<p>B = {jack, queen, ace}, probability of an event B = 12\/52.<\/p>\n<p>&nbsp;<\/p>\n<p>Favorable outcomes are 4, 12 on both event.<\/p>\n<p>&nbsp;<\/p>\n<p>b.\u00a0\u00a0<strong>Statistical approach<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Probability of happening an event E under a random experiment perform repeatedly under regular and identical circumstances is the ratio of the number of times event occurred to the number of\u00a0 trial. This\u00a0limit must be unique and finite.<\/p>\n<p>&nbsp;<\/p>\n<p>Probability of happening event E given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-172\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81.png\" alt=\"\" width=\"335\" height=\"57\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81.png 335w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81-300x51.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-81-225x38.png 225w\" sizes=\"auto, (max-width: 335px) 100vw, 335px\" \/><\/p>\n<p>This approach of probability is also known as <strong>Empirical<\/strong> approach of probability due it\u2019s repetitive behavior.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A survey conducted in Bathinda district of Punjab state to observe that how many people wearing helmets or not. Total population of Bathinda district is approx. 285813. Let the 40000 use regularly helmet. Probability of use of helmet on regular basis is 40000\/285813= 0.139951.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>c.<\/strong>\u00a0<strong>Subjective approach:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Subjective approach is completely biased. Numerical measure of this approach cannot be possible because it is based on biasedness, based on person\u2019s knowledge, beliefs, experience and intuition.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose one old person assumes the chance of cold in winter will be normal and not too cold. One business man observes the chance of no profit in the business this year.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Above example of subjective approach will vary person to person. This approach is not quite enough to measure probability numerically.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 6. Axioms of Probability<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Numerical measure of uncertainty to respective event is called probability. For accurate and best result of an event as much as possible probability must follow some axioms.<\/p>\n<p>&nbsp;<\/p>\n<p>Axioms of probability are given as:<\/p>\n<p>&nbsp;<\/p>\n<p><strong>a)\u00a0 Axioms of non-negativity:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u201cProbability of an event must be non-negative\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p>Suppose E is any event of a random experiment, probability of E is:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">?(?) \u2265 0<\/p>\n<p>&nbsp;<\/p>\n<p>Probability lies between one and zero, probability must be non-negative but less than one.<\/p>\n<p>&nbsp;<\/p>\n<p>0 \u2264 ?(?) \u2264 1.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>b)\u00a0 Axioms of certainty:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u201cProbability sum of all possible outcomes of a random experiment must be one\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p>Suppose E is an event of a random experiment, probability sum of all possible event is:<\/p>\n<p>&nbsp;<\/p>\n<p>?(?) = 1.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If probability of getting head at first time of tossing a coin is 0.5 and probability of getting tail on first time of tossing a coin is 0.5. Total of getting head or tail is 1.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>c)<\/strong>\u00a0\u00a0\u00a0 <strong>Axioms of additivity:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u201cProbability of union of all events is equal to the sum of the probability of all distinct event\u201d.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-173 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82.png\" alt=\"\" width=\"585\" height=\"112\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82.png 585w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82-300x57.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82-225x43.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-82-350x67.png 350w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">7. Permutations<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If a probability problem is <\/span>related<span style=\"text-align: initial;font-size: 1em\"> some specific type arrangement like <\/span>arrangement<span style=\"text-align: initial;font-size: 1em\"> of letter of a word, arrangement of persons in a queue etc. Permutations are used to solve <\/span>these type<span style=\"text-align: initial;font-size: 1em\"> of arrangement problem.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0\u00a0Mathematical form of permutation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Special arrangements of items defined by permutations. When selecting m items from the N items, possible permutations are given by this formula:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-174 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83.png\" alt=\"\" width=\"775\" height=\"215\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83.png 775w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83-300x83.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83-768x213.png 768w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83-65x18.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83-225x62.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-83-350x97.png 350w\" sizes=\"auto, (max-width: 775px) 100vw, 775px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>8. Combination<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If a probability problem is related to selection of some unit from a huge amount of same unit like selection a ball from a bag, selecting a card from a deck of card etc. Combinations are used to solve these type of selection problem.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Mathematical form of combination<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Selection of items from group of items defined by combination. When selecting m items from the N items, possible combination is given by:<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-175\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-84.png\" alt=\"\" width=\"166\" height=\"61\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-84.png 166w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-84-65x24.png 65w\" sizes=\"auto, (max-width: 166px) 100vw, 166px\" \/><\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Example<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-176 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85.png\" alt=\"\" width=\"607\" height=\"45\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85.png 607w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85-300x22.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85-65x5.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85-225x17.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-85-350x26.png 350w\" sizes=\"auto, (max-width: 607px) 100vw, 607px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>Possible combinations of selecting 2 marble from a bag of 6 marble = 2!4!6! = 15.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Question 1<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>One unbiased dice is thrown. Find the probability:<\/p>\n<p>&nbsp;<\/p>\n<p>a.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows 5<\/p>\n<p>b.\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows even number<\/p>\n<p>c.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Dice shows odd number<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-177\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-86.png\" alt=\"\" width=\"194\" height=\"101\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-86.png 194w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-86-65x34.png 65w\" sizes=\"auto, (max-width: 194px) 100vw, 194px\" \/><\/p>\n<p><strong>Answer<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">One unbiased dice is thrown randomly. Possible outcomes are 1, 2, 3, 4, 5 and 6. Mathematical approach is used to find probability for given event a, b, and c.<\/p>\n<p>&nbsp;<\/p>\n<p>For an event a:<\/p>\n<p>&nbsp;<\/p>\n<p>As\u00a0 dice\u00a0 shows\u00a0 five\u00a0 only\u00a0 one\u00a0 time.\u00a0 Favorable\u00a0 outcome\u00a0 is\u00a0 1\u00a0 and\u00a0 total\u00a0 possible\u00a0 outcomes\u00a0 are\u00a0 6.<\/p>\n<p>Probability of event a is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-178\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-87.png\" alt=\"\" width=\"189\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-87.png 189w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-87-65x19.png 65w\" sizes=\"auto, (max-width: 189px) 100vw, 189px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>For event b:<\/p>\n<p>&nbsp;<\/p>\n<p>As dice has three even number i.e. 2, 4, 6. Favorable outcomes are 3 and total outcomes are 6.<\/p>\n<p>Probability of event b is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-180\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-88.png\" alt=\"\" width=\"190\" height=\"59\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-88.png 190w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-88-65x20.png 65w\" sizes=\"auto, (max-width: 190px) 100vw, 190px\" \/><\/p>\n<p>For event c:<\/p>\n<p>&nbsp;<\/p>\n<p>Also dice has three odd number i.e. 1, 3, 5. Favorable outcomes are 3 and total outcomes are 6.<\/p>\n<p>Probability of event c is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-181\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-89.png\" alt=\"\" width=\"179\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-89.png 179w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-89-65x20.png 65w\" sizes=\"auto, (max-width: 179px) 100vw, 179px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Question 2<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">Suppose a coin is tossed two times.\u00a0 Find the probability of getting head one first time?<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Answer<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-182\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-90.png\" alt=\"\" width=\"177\" height=\"134\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-90.png 177w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-90-65x49.png 65w\" sizes=\"auto, (max-width: 177px) 100vw, 177px\" \/><\/p>\n<p>Coin has two possible outcomes one is head and second is tail.<\/p>\n<p>&nbsp;<\/p>\n<p>After tossing the coin two times. There are four possible outcomes<\/p>\n<p>&nbsp;<\/p>\n<p>i.e. HH, HT, TH, TT.<\/p>\n<p>&nbsp;<\/p>\n<p>Favorable outcomes for event are 2 (HH, HT) and total outcomes are 4.<\/p>\n<p>&nbsp;<\/p>\n<p>Probability of getting head on first place or on first coin is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-183\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-91.png\" alt=\"\" width=\"152\" height=\"49\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-91.png 152w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-91-150x49.png 150w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-91-65x21.png 65w\" sizes=\"auto, (max-width: 152px) 100vw, 152px\" \/><\/p>\n<p><strong>Question 3<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Suppose a bag has 20 with different colors.<\/p>\n<p>&nbsp;<\/p>\n<p>There are 5 green marbles, 6 blue marbles and 9 red marbles.<\/p>\n<p>&nbsp;<\/p>\n<p>Let us consider that two marbles are selected randomly.<\/p>\n<p>&nbsp;<\/p>\n<p>Find the probability of getting two red marbles, getting one red<\/p>\n<p>&nbsp;<\/p>\n<p>and one blue marbles, getting one blue and one green ball.<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-184\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-92.png\" alt=\"\" width=\"147\" height=\"178\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-92.png 147w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-92-65x79.png 65w\" sizes=\"auto, (max-width: 147px) 100vw, 147px\" \/><\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Answer<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Probability of getting two red marbles:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Bag contains 9 red marbles and total marbles are 20. We are interested in to find out the probability that two marbles are selected from 20 marbles and it must be red so these two selected marbles will be from 9 red marbles. Combination will be used to find out probability as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-185\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-93.png\" alt=\"\" width=\"120\" height=\"151\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-93.png 120w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-93-65x82.png 65w\" sizes=\"auto, (max-width: 120px) 100vw, 120px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-186\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-94.png\" alt=\"\" width=\"212\" height=\"55\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-94.png 212w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-94-65x17.png 65w\" sizes=\"auto, (max-width: 212px) 100vw, 212px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>As bag contains 9 red marbles and 6 blue marbles to find out the probability of one red marble and one blue marble. The formula is<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-187\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95.png\" alt=\"\" width=\"765\" height=\"502\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95.png 765w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95-300x197.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95-65x43.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95-225x148.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-95-350x230.png 350w\" sizes=\"auto, (max-width: 765px) 100vw, 765px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>9. Summary<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This module will help learner to understand basic of probability and terminology that is essential for better understanding of the problem. Probability approach is discusses that helps to transform the possibility into numerical results. In this module, main concern is only to introduce you keyconcepts and simple probability approach and application of probability. Advance probability problems will be discussed in the further probability module.<\/p>\n<\/div>\n<ol start=\"10\">\n<li><strong> Suggested Readings<\/strong><\/li>\n<\/ol>\n<p>Agresti, A. and B. Finlay, <em>Statistical Methods for the Social Science, 3<\/em><em>rd<\/em> <em>Edition, Prentice Hall<\/em>, 1997.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Daniel, W. W. and C. L. Cross, C. L., <em>Biostatistics<\/em>: <em>A Foundation for Analysis in the Health Sciences,<\/em> 10th Edition<em>,<\/em> John Wiley &amp; Sons, 2013.<\/p>\n<p>&nbsp;<\/p>\n<p>Hogg, R. V., J. Mckean and A. Craig, <em>Introduction to Mathematical Statistics<\/em>, Macmillan Pub. Co. Inc., 1978.<\/p>\n<p>&nbsp;<\/p>\n<p>Meyer, P. L., <em>Introductory Probability and Statistical Applications<\/em>, Oxford &amp; IBH Pub, 1975.<\/p>\n<p>&nbsp;<\/p>\n<p>Stephens, L. J., <em>Schaum\u2019s Series Outline: Beginning Statistics,<\/em> 2nd Edition<em>,<\/em> McGraw Hill, 2006.<\/p>\n<p>&nbsp;<\/p>\n<p>Triola, M. F., <em>Elementary Statistics<\/em>, 13th\u00a0 Edition, Pearson, 2017.<\/p>\n<p>&nbsp;<\/p>\n<p>Weiss, N. A., <em>Introductory Statistics<\/em>, 10th Edition, Pearson, 2017.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Introduction to Probability<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/u5sFDxoIvvo\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>One can refer to the following links for further understanding of the statistics terms.<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf\">http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html\">http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf\">http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/stats.oecd.org\/glossary\/\">https:\/\/stats.oecd.org\/glossary\/<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary\">http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm\">https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A\">https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A<\/a><\/p>\n","protected":false},"author":3,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-harmanpreet-singh-kapoor"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-162","chapter","type-chapter","status-publish","hentry","contributor-dr-harmanpreet-singh-kapoor"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/162","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":9,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions"}],"predecessor-version":[{"id":189,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions\/189"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/media?parent=162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapter-type?post=162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/contributor?post=162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/license?post=162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}