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The probability of India winning a test ...

The probability of India winning a test match againest England is `(2)/(3)`. Assuming independence from match to match, the probability that in a 7 match series India's third win occurs at the fifth match, is

A

`(8)/(27)`

B

`(16)/(81)`

C

`(8)/(81)`

D

`(32)/(81)`

Text Solution

Verified by Experts

The correct Answer is:
B
Consider the following events:
`A=` India wins 2 matches in first 4 matches.
`B=` India wins fifth match.
`C=` Getting any outcome in sixth and seventh matches.
We have,
`P(A)=.^4C_(2)((2)/(3))^2((1)/(3))^(4-2), P(B)=(2)/(3) and, P(C)=1`.
`therefore` Required Probability `P=(A cap B cap C)`
`=P(A)P(B)P(C)" "[because A,B,C "are independent"]` .
`=.^4C_(2)((2)/(3))^2((1)/(3))^2xx(2)/(3)xx 1=(16)/(81)`
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