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Question: The probability of India winning a test match against West Indies is 12. Assuming independence from ...

The probability of India winning a test match against West Indies is 12. Assuming independence from match to match the probability that in a match series India second win occurs at the third test is

A

1/6

B

1/4

C

1/2

D

2/3

Answer

1/4

Explanation

Solution

The problem asks for the probability that India's second win occurs at the third test.

Let P(W) be the probability that India wins a match, and P(L) be the probability that India loses a match.

Given P(W) = 1/2.

Since there are only two outcomes (win or lose), P(L) = 1 - P(W) = 1 - 1/2 = 1/2.

Matches are independent.

For the second win to occur at the third test, two conditions must be met:

  1. The third test must be a win (W).
  2. In the first two tests, India must have achieved exactly one win (and one loss).

Let's consider the possible sequences for the first three tests that satisfy these conditions:

  • Sequence 1: Win (1st), Loss (2nd), Win (3rd) - WLW

    The probability of this sequence is P(W) * P(L) * P(W) = (1/2) * (1/2) * (1/2) = 1/8.

  • Sequence 2: Loss (1st), Win (2nd), Win (3rd) - LWW

    The probability of this sequence is P(L) * P(W) * P(W) = (1/2) * (1/2) * (1/2) = 1/8.

These two sequences are mutually exclusive events. Therefore, the total probability that the second win occurs at the third test is the sum of their individual probabilities:

Required Probability = P(WLW) + P(LWW) = 1/8 + 1/8 = 2/8 = 1/4.

Alternatively, we can calculate the probability of getting exactly one win in the first two matches and then multiply it by the probability of winning the third match.

Probability of exactly one win in the first two matches:

There are C(2, 1) ways to get one win in two matches.

P(exactly one win in 2 matches) = C(2, 1) * P(W)^1 * P(L)^1 = 2 * (1/2)^1 * (1/2)^1 = 2 * (1/4) = 1/2.

The probability of winning the third match is P(W) = 1/2.

Since the matches are independent, the required probability is the product of these two probabilities:

Required Probability = P(exactly one win in first 2 matches) * P(win in 3rd match) = (1/2) * (1/2) = 1/4.