Question

A student appears for tests $I,II$ and $III$. The student is successful only if he passes either in tests $I$ and $II$ or tests $I$ and $III$(not all the $3$).
The probabilities of the student passing in tests $I,II,III$ are $p,q$ and $\frac{1}{3}$ respectively. The probability that the student is successful is $\frac{1}{6}$ and the relation between $p$ and $q$ is $\frac{1}{k}=pq+p$. Then the value of $k$ is:

Answer 1

Probability that the student is successful $=\frac{1}{6}$
$=$ Probability that student passes only in tests $I$ & $II$ or tests $I$ & $III$ $=P\left(I\right)P\left(II\right)P\left(II{I}^{\prime }\right)+P\left(I\right)P\left(I{I}^{\prime }\right)P\left(III\right)=\left(p\right)\left(q\right)\left(\frac{2}{3}\right)+\left(p\right)(1-q)\left(\frac{1}{3}\right)=\frac{pq}{3}+\frac{p}{3}$
$\Rightarrow \frac{1}{6}=\frac{pq}{3}+\frac{p}{3}\Rightarrow \frac{1}{2}=pq+p$
Since, $\frac{1}{k}=pq+p$
Hence, $k=2$