10 Introduction to Probability
Dr. Harmanpreet Singh Kapoor
Learning Objectives
- Introduction
- Terminology
- Types of Events
- Probability Approach
- Axioms of Probability
- Permutations
- Combination
- Summary
- Suggested Readings
1. Learning Objectives
The objective of this module is to introduce key concepts that help to understand probability problem in simple words and different approaches to find numerical measure of probability for different probability problem. Examples and some images are used for better explanation of this topic in an easy manner.
2. Introduction
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.
According to Ya-Lin-Chou “Probability is the science of decision making with calculated risks in the face of uncertainty”.
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.
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.
3. Terminology
There is a need to be aware about some basic concepts of probability before going to numerical measure of probability. In this section, focus will be on key concepts to understand a probability problem in words and convert into numerical form to solve it by using different approach of probability according to the features of the problems.
Key concepts are given as:
3.1 Random Experiment
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.
“Random 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.”
List of some examples of random experiment is given below:
- Tossing a coin.
- Rolling a dice.
- Observe risk of an investor.
- Selecting a card from a deck of card.
- Selecting a ball from a group of balls.
- Arrangement of persons in a queue.
- Observe the number of Xiaomi android phones sold by Amazon in 2017.
- Selecting a student from a classroom.
- Observe the number of wickets in a cricket match.
- Observe product of a process in a manufacturing company.
- Observe a bulb is either fuse or not.
- Observe a person who speaks either truth or lie.
- Observe the number of car accident in particular place.
- Observe the number of birth in India in year 2017.
- Observe the share market prices.
3.2 Outcome
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.
“Outcome is a result of a random experiment”.
List of some outcomes of random experiment is given as:
- Outcome of tossing a coin: head (H) or tail (T).
- Outcome of rolling a dice: 1/ 2/ 3/ 4/ 5/ 6.
- Outcome of selecting a ball from a group of different colors balls like blue, red, white: blue/ red/ white.
- Outcome of risk of an investor: high risk/ neutral risk/ low risk/ no risk.
- Outcome of selecting a card from a deck of card: 1/ 2/ 3/…. /10/ J/ Q/ K/ A/ club/ spade/ diamond/ heart/ red/ black.
- Outcome of working condition of a bulb: Fuse/not fuse. Outcome of a person speaking tendency; truth/ lie.
- Outcome of observing car accident in a state: 1, 2, ……
- Outcome of product quality in a manufacturing company: defective/ non-defective.
3.3 Sample space:
After identifying the result of the experiment, there is a need to arrange all possible outcome. In basic mathematics, the definition of set was described. Set is a group of well-defined objects that related to each other under same consideration. Here, result of an experiment is well defined different result under same consideration, so set is used as sample space to arrange all possible outcomes.
“Sample space is set of all possible outcomes of an experiment”.
List of some examples of sample space are:
- Sample space for tossing a coin: {H, T}.
- Sample space for rolling a dice: {1, 2, 3, 4, 5, 6}.
- Sample space for selecting a card from a deck of card: {1, 2, …, 10, A, K, Q, J}.
- Sample space for selecting a ball from a group of balls of different color like blue, green and red: {Blue ball, red ball, green ball}.
- Sample space of observe risk of an investor: {high, low, neutral, no}.
- Sample space of product that produces by a manufacturing company: {defective, non-defective}.
- Sample space of observe a bulb: {fuse, non-fuse}.
- Sample space of observing person talk: {truth, lie}.
- Sample space for observe car accident in a state: {1, 2, 3…}.
- Sample space for observe the number of wickets in a cricket match: {1, 2, …, 22}.
3.4 Event
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.
“Event is a particular performance of a random experiment”.
List of some examples of event:
- Suppose a random experiment is tossing a coin two times.
Sample space: {HH, HT, TH, TT}
Possible event: getting head both times, getting tail both times, getting one head and tail of a coin.
- Suppose rolling a die two times is a random Experiment.
Sample space: {(1,1), (1,2),………,(1,6), (2,1), (2,2), ……..,(2,6), (3,1), (3,2),…….,(3,6), (4,1), (4,2), ……..,(4,6), (5,1), (5,2),…….,(5,6), (6,1), (6,2),…..,(6,6)}.
Possible event: Getting same number both times , getting odd number at first time and even number at second time, getting odd number at first time, getting sum of both is even , getting sum of both time is odd , many more event may occur.
- Selecting a card from a deck of cards is an random experiment.
Sample space: {1, 2, ……,10, A, K, Q, J}.
Possible event: 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.
4. Types of Events
There are many types of event according to the happening of event and outcome. In this section, types of event are defined.
4.1 Simple event:
Suppose a particular event happens among all possible events. Result of particular event is at most one, then event is called simple event.
“An event that shows exactly one outcome of a random experiment is a simple event”.
Example
Suppose a quality analyst analysis two processes’ product in a random experiment and classify the process product as defective product and non-defective product.
Sample space: {nn, nd, dn, dd}
where d is for defective product and n is for non-defective product.
Let E is an event.
Event E : neither product is defective.
E ={nn}; as E has only one outcome, E is a simple event.
Example
Tossing a coin two times in a random experiment.
Sample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail
Let E be an event: getting head both time.
E = {HH}; E has only one outcome, E is a simple event.
4.2 Compound event
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.
“A event shows more than one outcome of a random experiment is a compound event”.
Example
Suppose a quality analyst analysis two processes’ product quality in an random experiment and classify the process product as defective product and non-defective product.
Sample space : {nn, nd, dn, dd}
where: d is for defective product and n is for non-defective product.
Let E is an event.
Event E : at least one product is defective.
E: {nd, dn, dd}; E has more than one outcome. So, E is a compound event.
Example
Tossing a coin two times is a random experiment.
Sample space: {HH, HT, TH, TT}; where H: represents head, T: represents tail
Let E be an event getting head at least one time.
E: {HH, HT, TH}; E has more than one outcome. So, E is a compound event.
4.3 Exhaustive event
All possible results of an experiment is termed as an exhaustive event.
“Exhaustive event is the total outcome of a random experiment”.
List of some examples are:
- Tossing of one coin is a random experiment, exhaustive cases are 2.
- Throwing a dice, exhaustive cases are 6.
- Throwing two dices, exhaustive cases are 62.
- Tossing two coins, exhaustive cases are 22.
- Selecting a card, exhaustive cases are 52.
4.4 Dependent event
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.
“Event is said to be dependent event if happening of one event in one trail is affected by happening of similar event in next trail”.
Examples:
- 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.
- 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.
4.5 Favorable event
Outcomes of a particular event are favorable to that events. These outcomes of events are considered as favorable event.
“Favorable event is the number of outcome favor to an event”.
Examples
- Tossing two coins is a random experiment and getting head on both coins is an event. Exhaustive event is 22 = 4 and favorable event is 1 (i.e. head and head).
- Throwing two dice is an random experiment and getting same number on both side is an event, favorable event is 6 {(1,1), (2,2), (3,3), (4,4), ( 5,5), (6,6)}.
4.6 Mutually exclusive event
“Events can’t occur simultaneously in same trial are mutually exclusive events”.
Suppose A and B are two events. If A and B are mutually exclusive event, their venn diagram is given in Figure 2.
Figure 2
Examples
- 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.
- 1, 2, 3, 4, 5, 6 are the possible six outcome of random experiment in throwing a dice. All of these outcomes cannot occur in a single trail, so they are all mutually exclusive event.
- A switch cannot be on/ off at same 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 event cannot occur at same time, so A and B are mutually exclusive event.
4.7 Independent event
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.
“Event 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”.
Examples
- 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.
- Tossing a coin is a random experiment, if one get 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 independent event.
- Drawing a card from a pack of card with replacement is independent event. Every time total number of cards remain same.
4.8 Equally likely events
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.
“Equally likely events shows equal probability for all events. There is no preference for any event, all have equal preference”.
Example
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.
Probability of getting head is ½.
Probability of getting tail is ½.
Both have equal probability and equal preference. So, events are equally likely events.
Example
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.
4.9 Impossible event
If an event under a random experiment cannot occur, that event is known as an impossible event.
“Numerical measure of an event is zero then the event is impossible event”.
For an impossible event E,
P(E) = 0.
Example
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.
4.10 Certain event
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.
“Numerical measure of an event is one, then event is certain event”.
For a sure and certain event E:
P(E) = 1.
Example
Hitting a dart is a random experiment and throwing a dart at center 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 sure event.
4.11 Complementary event
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.
?(?) + ?(?? ) = 1.
Example
- 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.
- Rolling a dice is a random experiment and getting 6 on dice is event. Getting value other then 6 or not getting 6 is non-happening event and complementary event.
5. Probability Approach
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.
a. Mathematical approach
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.
If “F” denotes the favorable outcome of an event (E) of a random experiment and “T” denotes the total possible outcome. The probability of happening an event (E) is given by:
Note: All possible outcomes in mathematical approach must be exhaustive, mutually exclusive and equally likely.
Example
Suppose a random experiment of randomly selecting a card from a deck of cards of 52 has equally likely outcomes.
Let event A = {king}, probability of an event A = 4/52
B = {jack, queen, ace}, probability of an event B = 12/52.
Favorable outcomes are 4, 12 on both event.
b. Statistical approach
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 trial. This limit must be unique and finite.
Probability of happening event E given by:
This approach of probability is also known as Empirical approach of probability due it’s repetitive behavior.
Example
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.
c. Subjective approach:
Subjective approach is completely biased. Numerical measure of this approach cannot be possible because it is based on biasedness, based on person’s knowledge, beliefs, experience and intuition.
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.
Above example of subjective approach will vary person to person. This approach is not quite enough to measure probability numerically.
6. Axioms of Probability
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.
Axioms of probability are given as:
a) Axioms of non-negativity:
“Probability of an event must be non-negative”.
Suppose E is any event of a random experiment, probability of E is:
?(?) ≥ 0
Probability lies between one and zero, probability must be non-negative but less than one.
0 ≤ ?(?) ≤ 1.
b) Axioms of certainty:
“Probability sum of all possible outcomes of a random experiment must be one”.
Suppose E is an event of a random experiment, probability sum of all possible event is:
?(?) = 1.
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.
c) Axioms of additivity:
“Probability of union of all events is equal to the sum of the probability of all distinct event”.
7. Permutations
If a probability problem is related some specific type arrangement like arrangement of letter of a word, arrangement of persons in a queue etc. Permutations are used to solve these type of arrangement problem.
Mathematical form of permutation
Special arrangements of items defined by permutations. When selecting m items from the N items, possible permutations are given by this formula:
8. Combination
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.
Mathematical form of combination
Selection of items from group of items defined by combination. When selecting m items from the N items, possible combination is given by:
Example
Possible combinations of selecting 2 marble from a bag of 6 marble = 2!4!6! = 15.
Question 1
One unbiased dice is thrown. Find the probability:
a. Dice shows 5
b. Dice shows even number
c. Dice shows odd number
Answer
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.
For an event a:
As dice shows five only one time. Favorable outcome is 1 and total possible outcomes are 6.
Probability of event a is given by:
For event b:
As dice has three even number i.e. 2, 4, 6. Favorable outcomes are 3 and total outcomes are 6.
Probability of event b is given by:
For event c:
Also dice has three odd number i.e. 1, 3, 5. Favorable outcomes are 3 and total outcomes are 6.
Probability of event c is given by:
Question 2
Suppose a coin is tossed two times. Find the probability of getting head one first time?
Answer
Coin has two possible outcomes one is head and second is tail.
After tossing the coin two times. There are four possible outcomes
i.e. HH, HT, TH, TT.
Favorable outcomes for event are 2 (HH, HT) and total outcomes are 4.
Probability of getting head on first place or on first coin is
Question 3
Suppose a bag has 20 with different colors.
There are 5 green marbles, 6 blue marbles and 9 red marbles.
Let us consider that two marbles are selected randomly.
Find the probability of getting two red marbles, getting one red
and one blue marbles, getting one blue and one green ball.
Answer
Probability of getting two red marbles:
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:
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
9. Summary
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.
- Suggested Readings
Agresti, A. and B. Finlay, Statistical Methods for the Social Science, 3rd Edition, Prentice Hall, 1997.
Daniel, W. W. and C. L. Cross, C. L., Biostatistics: A Foundation for Analysis in the Health Sciences, 10th Edition, John Wiley & Sons, 2013.
Hogg, R. V., J. Mckean and A. Craig, Introduction to Mathematical Statistics, Macmillan Pub. Co. Inc., 1978.
Meyer, P. L., Introductory Probability and Statistical Applications, Oxford & IBH Pub, 1975.
Stephens, L. J., Schaum’s Series Outline: Beginning Statistics, 2nd Edition, McGraw Hill, 2006.
Triola, M. F., Elementary Statistics, 13th Edition, Pearson, 2017.
Weiss, N. A., Introductory Statistics, 10th Edition, Pearson, 2017.
| you can view video on Introduction to Probability | |
One can refer to the following links for further understanding of the statistics terms.
http://biostat.mc.vanderbilt.edu/wiki/pub/Main/ClinStat/glossary.pdf
http://www.stats.gla.ac.uk/steps/glossary/alphabet.html
http://www.reading.ac.uk/ssc/resources/Docs/Statistical_Glossary.pdf
https://stats.oecd.org/glossary/
http://www.statsoft.com/Textbook/Statistics-Glossary
https://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm