Classical Definition of Probability and Its Limitations

We assume that the total number of outcomes of the random experiment is finite, say N. We also assume all the outcomes are equally likely to occur, i.e. it is assumed that the circumstances of the experiment are such that no outcome is more likely to occur than the other. Under these assumptions if we have n(B) outcome/ elementary events in favour of a particular event B then probability of the vent B is,

P(B)= n(B)/N.

As N is the total number of possible outcomes of the random experiment thus, n(B) can not exceeds N.

n(B) ≤ N , therefore P(B) ≤ 1.

Examples :

Two unbiased coins are tossed. What is the probability of getting at least one head?

The possible cases are HH, HT, TH, TT.

Thus the sample space, S = {HH, HT, TH, TT}.

All these cases are equally likely. Thus n(S) = 4 (=N). The event B (at least one head) occurs if either of HH, HT, TH occurs.

We can construct a set with the outcomes in favour of the event B as follows;

B = {HH, HT, TH}

Thus P(B) = n(B)/ n(s) = 3/4

Game of dice of Mhabharat :

Suppose in the famous game of dice of Mahabharata, Shakuni decides to roll two unbiased dices and he offers Yudhisthir that if the product of the numbers appears in two dices are odd then Yudhisthir will win other wise Shakuni will win.

Now Yudhisthir thinks with his simple mind that the product of two integers is either even or odd thus his sample space is, Sy= { Even , Odd}.

By classical definition of probability he thinks probability of his win is ½ no less than the probability of Shakuni. To, him the game is fair.

Shakuni thinks of the game more meticulously. He consider the sample space of the throwing two dice, unlike Yudhisthir (Yudhistir considered the product of two randomly generate numbers between 1 to 6 as random experiment). Thus his sample space is ,

Sk = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

As the dices are unbiased, all the 36 outcomes are equally likely. Shakuni knows that there are only 9 outcomes out of 36, where the product is odd. Thus the elements of the sample space of Yudhisthir are not equally likely and we all the result of the game.

Another example:

Suppose you have two big containers. Container 1 contains 5 red balls and 5 white balls and Container 2 contains 8 red balls and 2 white balls.

You will draw a ball randomly from Container 1. What is the probability of getting a red ball?

Your friend will draw a ball from the Container 2. What is the probability of getting a red ball from Container 2?

•Hint: Suppose both of you have number stickers from 1 to 10. Use them to clear your concept.