Probability-9 (IIT-1995)

The probability of India winning a test match against West Indies is ½. Assuming independence from match to match the probability that in a 5 match series India’s second win occurs at the third test is:  

(a) 1/8          (b) 1/4        (c) 1/2              (d) 2/3

Solution follows here:

Solution:

The probability of India winning a test match against West Indies = ½

=> The probability of India loosing a test match against West Indies =1- ½ = ½

The criteria here is “India’s second win should occur at the third test”

The following table summarizes all the possibilities:

	1st Match	2nd Match	3rd Match	P(E) = P(E1)*P(E2)*P(E3)
India loosing all of the first three	L	L	L	1/2*1/2*1/2 = 1/8
India winning one of the first three	W	L	L	1/2*1/2*1/2 = 1/8
	L	W	L	1/2*1/2*1/2 = 1/8
	L	L	W	1/2*1/2*1/2 = 1/8
India winning two of the first three	L	W	W	1/2*1/2*1/2 = 1/8
	W	L	W	1/2*1/2*1/2 = 1/8
	W	W	L	1/2*1/2*1/2 = 1/8
India winning all of the first three	W	W	W	1/2*1/2*1/2 = 1/8

Out of all the 8 possibilities, the desired possibilities are two only (shown in green colour)

Probability     =1/8 + 1/8

= 2/8 = 1/4

Answer (b)