The probability of India winning a test match against Westindies is \(

1.	The probability of India winning a test match against Westindies is \(\frac{1}{2}\). Assuming independence from match to match, the probability that in a match series, India’s second win occurs at the third test is (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{4}\) (d) \(\frac{2}{5}\)
Answer» (c) \(\frac{1}{4}\) Let A = Event that India wins the match. Then, \(\bar{A}\) = Event that India loses the match. P(A) = \(\frac{1}{2}\) and P(\(\bar{A}\)) = 1 - \(\frac{1}{2}\) = \(\frac{1}{2}\) (∵ P(A) + P(\(\bar{A}\)) = 1) P(Third match is India’s second winning match) = P(India wins the 1^st match, loses 2^nd match, wins 3^rd match) + P(India loses 1^st match, wins 2^nd match, wins 3^rd match) (or) ⇒ Required probability = P(A \(\bar{A}\) A) + P( \(\bar{A}\) A A) = P(A) x P(\(\bar{A}\)) x P(A) + P(\(\bar{A}\)) x P(A) x P(A) = \(\frac{1}{2}\) x \(\frac{1}{2}\) x \(\frac{1}{2}\) + \(\frac{1}{2}\) x \(\frac{1}{2}\) x \(\frac{1}{2}\) = \(\frac{1}{8}\) + \(\frac{1}{8}\) = \(\frac{2}{8}\) = \(\frac{1}{4}\).

The probability of India winning a test match against Westindies is \(\frac{1}{2}\). Assuming independence from match to match, the probability that in a match series, India’s second win occurs at the third test is (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{4}\) (d) \(\frac{2}{5}\)

Answer»

(c) \(\frac{1}{4}\)

Let A = Event that India wins the match. Then,

\(\bar{A}\) = Event that India loses the match.

P(A) = \(\frac{1}{2}\) and P(\(\bar{A}\)) = 1 - \(\frac{1}{2}\) = \(\frac{1}{2}\) (∵ P(A) + P(\(\bar{A}\)) = 1)

P(Third match is India’s second winning match)

= P(India wins the 1^st match, loses 2^nd match, wins 3^rd match) + P(India loses 1^st match, wins 2^nd match, wins 3^rd match)

(or)

⇒ Required probability = P(A \(\bar{A}\) A) + P( \(\bar{A}\) A A)

= P(A) x P(\(\bar{A}\)) x P(A) + P(\(\bar{A}\)) x P(A) x P(A)

= \(\frac{1}{2}\) x \(\frac{1}{2}\) x \(\frac{1}{2}\) + \(\frac{1}{2}\) x \(\frac{1}{2}\) x \(\frac{1}{2}\) = \(\frac{1}{8}\) + \(\frac{1}{8}\) = \(\frac{2}{8}\) = \(\frac{1}{4}\).

The probability of India winning a test match against Westindies is \(\frac{1}{2}\). Assuming independence from match to match, the probability that in a match series, India’s second win occurs at the third test is (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{4}\) (d) \(\frac{2}{5}\)

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