Question

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 atleast two tests. The probability that the candidate is selected, is

• A. $p^{2}(2-p)$
• B. $p(2-p)$
• C. $p+p^{2}+p^{3}$
• D. $p^{2}(1-p)$

Answer 1

Answer: (A)

Required probability
$=$ Probability of passing two test
$+$ Probability of passing all three test
$=P$ (passing I and II tests and fail in third test)
$+P$ (passing Ist test, fail in II test and passing in Illrd test)
$+P$ (fail in I test, passing in Ilnd and IIlrd tests)
$+P$ (passing in all three tests)
$=n p q+p q \times \frac{p}{2}+q \frac{p}{2} \times p+p p p$
$=p^{2}(1-p)+p(1-p) \frac{p}{2}+(1-p) \frac{p^{2}}{2}+p^{3}$
$=p^{2}\left(1-p+\frac{1}{2}-\frac{p}{2}+\frac{1}{2}-\frac{p}{2}+p\right)=p^{2}(2-p)$