9 Probability and Probability Distribution Functions

Prof. Aslam Mahmood

1) E-Text

Probability

Frequency distribution tables give us information about an event either in space or in time. A frequency distribution table is a compact form of information for my background knowledge about a particular phenomenon and help us in perceiving future course of the particular event. For example if we know the pattern of rainfall in a particular area for last 100 years, we can make the judgment about its occurrence next year with some confidence. Similarly, if we know the agricultural productivity of farms in a region for several years, we can generate the productivity distribution for coming years also.

Frequency distribution tables which gives us the number of times a particular event occurs out of its several outcomes are quite helpful in this regards. For example if we take the old record of the results of the total of 50 matches between two football teams A and B it may give us the following result:

S.N.	Result	Number	%
1	Team A will win	25	50
2	Team B will win	15	30
3	Draw	10	20
	Total	50	100

  However, before the match is played, if we want to know the result, we can only guess it on the basis of the past performance of two teams. In the present case we can say that fifty percent chances are that the team A will win thirty percent chances are in favour of team B and there are only twenty percent chances that neither team A nor team B will win. If our guess is base on very small number of cases we will not be very sure. On the other hand if we have information on very large number of matches we can make more reliable guess. In geography and in other social and natural sciences also we have to make certain judgment about an event for which we are not very sure. In such, cases we evaluate the reliability of judgments and choose the one which has high reliability. The reliability of any such judgment depends on our past knowledge of similar events and their outcomes. The possible outcomes will give us some idea about its occurrence and will be known as the probability of its occurrence. Theory of probability is developed on the basis of the logic of the occurrences of events and applying mathematical logic to measure them.

     Theory of probability deals with measurement of chances of the occurrence of an event on the basis of the basic characteristics of nature of its occurrence. It can also be measured empirically on the basis of the past record of its occurrence. In its simplest form , if an event can occur in n number of mutually exclusive and exhaustive random ways out of which m number of events are in favour of a particular event “ A “ then the probability of the occurrence of event “ A” is defined as p(A) = m/n provided n is sufficiently large.Here mutually exclusive means that if one event occurs other events of the same set cannot occur. Also exhaustive means total n number of ways include all possible ways the event can occur, no event is left in counting.

Example

From a deck of playing cards one card can be randomly drawn in 52 different ways. Out of which each set will have 13 ways to be drawn. Thus the probability that if a card is drawn at random it will belong to any set, say, diamond is 13/52= 1/4. Similarly the probability that it will belong to any other set is also 13/52=1/4. Similarly there are 4 cards of each denominations in all the set and the probability of each denomination is 4/52 =1/13

Axioms(definitional Properties) of probability

1. Sum of the probabilities of all the events is equal to 1.

2. Probability of not occurrence of an event = 1 – probability of its occurrence.

3. Sum of the probabilities of all possible events is equal to 1.

4. Minimum value of a probability is zero associated with impossible event (which can never occur).

5. Maximum value of probability is one associated with sure events (which will always occur).

Probability of Compound Events

Joint occurrence of two or more events is also possible. These events can occur in and/or form. Like we can think two events A and B occurring in form of either A will occur or B will occur( or both will occur if they can occur jointly). Another possibility is that event and B will occur jointly. There are different laws which will govern the probability of these events. The first will be governed by what is known as: “Additive Law of Probability” and the second will be governed by “ Multiplicative Law of Probability”.

   Additive Law of Probability

If the probability of the occurrence of two events A and B are P1and P2respectively then the probability that either A or B will occur will be:

P ( A + B ) = P (A ) + P ( B ) – Probability of their joint occurrence = P1+ P2 – P1* P2

Multiplication Law of Probability

There are some events which can occur simultaneously also. The probability of such events is the product of their individual probabilities and is represented by P ( A*B ). Thus ,

P ( A X B ) = p1*p2.

P ( A X B ) = 0 that is the events can not occur jointly. Such events are known as mutually exclusive events.

Example

From a deck of playing cards, a card is randomly drawn. Give the probability that it is:

( a )       Either a spate ( A ) or an ace ( B ).

( b )       Either a Spate or a Diamond ( C ).

( a ):      P1 , the probability of a card being Spate is 13/52.

P2, the probability of a card being an ace is 4/52.

Probability of the card being an ace from spate

P ( A X B ) = p1*p2., the probability of a card being spate X probability of a card being an ace is (13/52)* (4/52) = 1/52.

Probability of the Card being either a spate or an ace .

P ( A + B ) =P1 + P2 -p1*p2 = 13/52 + 4/52 – 1/52 = 16 /52.

( b ):     P1 , the probability of a card being Spate is 13/52.

P3 , the probability of a card being Diamond is 13/52.

Probability of the Card being either a spate or a diamond.

P ( A + C ) = P ( A ) + P ( C ) = 13/52 + 13/52 = 2/4= 1/2 ( Note that since a card can be either a Spate or a Diamond, it can not be both the probability of their joint occurrence will be zero i.e. p1 *p3 = 0 ).

Probability Distributions

The nature of the frequency distribution of any variable depends on the particular process which will generate it. For example the distribution of size of the land holdings of an area will be skewed to the right and will depend on the probability of the size of holdings of an individual which will decrease as the size will increase. The distribution of the average annual rainfall of an area over large number of years will be symmetrical i.e. it will be around average in most of the years and equally increase and decrease as we move from either side of the average.

Similarly, if four coins are tossed together, the results can be tabulated into five classes/ categories of events i.e. no head, one head, two heads, three heads and four heads. The probability of their occurrence will not be the same; each event will have a different probability which can be worked out empirically by tossing the four coins together for large number of time say ten thousand time or more and generating the frequency distribution table. Proportion of the number of times each event occur to the total number of times four coins are tossed will give the “empirical” probability of each event and is known as Probability Distribution. Once we know the probability distribution of any event, we can make predict the estimated value of all the possible events by multiplying the probability with the corresponding event. For example, if the probability of the average annual rainfall of an area being between 50 to 60cms. is 0.20, in coming 10 years the estimated number of years with average annual rainfall between 50 to 60 cms. will be 0.20 x 10 = 2 years only. We can also work out estimated numbers of years with any average annual rainfall, if we know its probability.Theory of probability has its greater applicability in theory of sampling and inferences. In any exercise of sampling we try to generalize about the universe with the help of findings of the sample. These generalizations are never 100% accurate. There is certain amount of uncertainty involved in it. In such situations we make probabilistic generalizations and specify the level of uncertainty. Theory of probability also help in simulation excercises.

Avoiding the long process of tossing the coins we can also generate the probability distribution using the properties of the theory of probability for the above process. Functional form of relationship between the values of a variable and their probabilities is known as Probability Distribution Function and it will be depending on the underlying assumptions behind the process which will generate it. Some of the important probability distributions are discussed below.

Binomial Distribution

In the example of tossing of coins, if we take a general form of n coins and make the following assumptions :

Probability of head is P Probability of tail is Q

The variable( x ) i.e. number of heads ( or tails ) is a discontinuous variable Occurrence of one coin is independent of the occurrence of other coins Number of coins ( n ) is not very large

Neither P nor Q is very small

We can generate the probabilities of number of heads being one , two, three, four ,…… and n. as

Where n is the number of coins, r is the number of heads and

( −1)( −2)…..1

_____________

(  −1)(  −2)…1( −  )( −  −1)…1

Cnx in fact is the number of possible combinations of X number of objects which can be made out of n given objects. For example if we have 4 objects a,b,c and d, combinations of 2 are ab, ac, ad, bc, bdand dc and if instead of actually making the combinations we are interested only in the number of such combination we can know it through the formula of permutations and combinations Cnx. which uses factorial numbers.

A number when multiplied by all the numbers reducing by one each time until we reach 1 is known as factorial number and factorial 0 is 1. A factorial number is denoted by the sign ( ! ). Thus ; 4! = 4X3X2x1 = 24 or 8! = 8x7x6x5x4x3x2x1= 40320 or n! = n (n-1) (N-2) (n-3)……..1.

Substituting different values of n and r we can work out different probabilities like P1 P2 P3P4 and P5 etc. The above function given in (1) is known as Binomial Distributionfunctionsince there are only two options.

Example

If four coins are tossed and the probability of head=probability of tail=1/2 we can work out the probabilities as below:

The count of different number of heads can also be plotted on a graph as shown below:

Count of number of heads

Thus theoretical Probability Distribution Functions give us the probabilities of different events without their actual trails, provided we can specify the model.

Properties of the Binomial Distribution

1.      It is symmetrical when p=q.

2.      If p≠ q, the Binomial distribution will be skewed.

3.      Mean of the Binomial distribution = np

4.      Standard deviation of a Binomial distribution = √npq.

Binomial distribution is a symmetrical distribution. We can see that the probability is as low as 1/16 for zero head. As it increases to one head it rises to 4/16 and for two heads it goes to maximum of 6/16, for three heads it decline to again 4/16 and finally reaches the initial value for four heads as 1/16. One can verify that the sum of the probabilities of all these exhaustive events is one.

Poisson Distribution

The Binomial distribution assumes a convenient form in the limiting case when:

1.      Either p or q is very small.

2.      Number of observations n is very large.

3.      The average value np is finite.

The probability distribution Pr will become as given below and is known as Poisson distribution. Pr =( −  mx )/X! where m is the mean of the variable X , X is the value of the variable e = 2.718… is a mathematical constant Poisson distribution in which either p or q is very small is suitable to generate the probabilities of rare events i.e. as the number of the variable becomes large their occurrence become smaller and smaller like number of floods in a river over say in 100 years or distribution of size of land holdings among the farmers of a district etc.

Properties of Poisson distribution:

1. Mode of the poisson distribution is the integral value between m and m-1.
2. Mean and standard deviation of a Poisson distribution are same and are equal to m.
3. Poisson distribution is skewed, not symmetrical.

   Normal Distribution

The above mentioned probability distributions i.e. Binomial distribution and Poisson distribution both are discontinuous distributions. If we have a continuous variable and the probability of the occurrence of the event is not very low and n is large, the “Binomial’ distribution takes a limiting form known as “Normal’ distribution.

Characteristics and uses of Normal distributions are discussed separately in a different module.

References

• Harvey D. (1969):Explanation in Geography, Arnold Publishers.
• Mood, A.M. and Graybull F.A. (1963) : Introduction to the Theory of Statistics, McGraw Hills
• Mahmood Aslam (1998): Statistical Methods in Geographical Stusies, Rajesh Publications New Delhi.