INTRODUCTION TO STATISTICS AND ECONOMETRICS

COUNTING TECHNIQUES

1.2.1 Simple Events with Equal Probabilities

Axiom (3) suggests that often the easiest way to calculate the probability of a composite event is to sum the probabilities of all the simple events that constitute it. The calculation is especially easy when the sample space consists of a finite number of simple events with equal probabilities, a situation which often occurs in practice. Let n(A) be the number of the simple events contained in subset A of the sample space 5. Then we have

n(A) n(S) '

Two examples of this rule follow.

example 2.3.1 What is the probability that an even number will show in a roll of a fair die?

We have n(S) = 6; A = {2, 4, 6) and hence n(A) = 3. Therefore, P(A) = 0.5.

example 2.3.2 A pair of fair dice are rolled once. Compute the prob­ability that the sum of the two numbers is equal to each of the integers 2 through 12.

Let the ordered pair (i, j) represent the event that і shows on the first die and у on the second. Then S = ((i, j) | i, j = 1, 2,... , 6), so that n(S) = 36. We have

n(i + j = 2) = »[( 1,1)] = 1, n(i+j = 3) = n[(l, 2), (2, 1)] = 2, n(i + j = 4) = n[(l, 3), (3, 1), (2, 2)] = 3, and so on. See Exercise 2.