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A candidate takes three tests in succession and the probability of passing the first test is p. The probability of passing each succeeding test is p or $\frac{p}{2}$ according as he passes (Or) fails in the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is

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$p + p^{2} + p^{3}$

$p^{2} (2 - p)$

$p^{2} (1 - p)$

$p (2 - p)$

answer is D.

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Detailed Solution

Let $A_{1}, A_{2}, A_{3}$ be the events that the candidate passes 1^st test,2^nd, 3^rd test respectively. Req. Prob $= P (A_{1} A_{2} A_{3} \cup A_{1} A_{2} {\bar{A}}_{3} \cup A_{1} {\bar{A}}_{2} A_{3} \cup {\bar{A}}_{1} A_{2} A_{3})$

$= P^{3} + P^{2} (1 - P) + P (1 - P) (\frac{P}{2}) + (1 - P) (\frac{P}{2}) (P) = P^{2} (2 - P)$

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