Question

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the $\text{1st}$ exam is $p$. If he fails in one of the exams, then the probabiliy of his passing in the next exam is $\frac{p}{2}$, otherwise it remains the same. Then the probability that he will qualify, is

• A. $2p^2-p$
• B. $2p^2-2p$
• C. $2p^2-p^3$
• D. $2p^2+p^3$

Answer 1

The correct option is C $2p^2-p^3$

Let $E_1,E_2,E_3$ be the events of passing $1^{st},2^{nd},3^{rd}$ tests respectively.
$E:$ denotes student will qualify
Probability of passing atleast two tests
$P\left(E\right)=\left[P\right(E_1)\times P(E_2/E_1\left)\right]+\left[P\right(E_1)\times P(E_2^{\prime }/E_1)\times P(E_3/E_2^{\prime }\left)\right]+\left[P\right(E_1^{\prime })\times P(E_2/E_1^{\prime })\times P(E_3/E_2\left)\right]$
$=p\cdot p+p\cdot (1-p)\cdot \frac{p}{2}+(1-p)\cdot \frac{p}{2}\cdot p$
$=p^2+\frac{p^2}{2}-\frac{p^3}{2}+\frac{p^2}{2}-\frac{p^3}{2}$
$=2p^2-p^3$