Question

A student appears for tests $I,II$ and $III$. The student is successful if he passes either in tests $I$ and $II$ or tests $I$ and $III$. The probabilities of the student passing in tests $I,II,III$ are $p,q$ and $1/2$ respectively. The probability that the student is successful is $1/2$ and the relation between $p$ and $q$ is $\frac{1}{p}=k+2q$. Then find the value of $k$.

Answer 1

Probability that the student is successful $=\frac{1}{2}=$ probability that student passes exam $I$&$II$ or exam $I$&$III$ $=P\left(I\right)P\left(II\right)+P\left(I\right)P\left(III\right)=\left(p\right)\left(q\right)+\left(p\right)\left(\frac{1}{2}\right)=pq+\frac{p}{2}$
$\Rightarrow$ $\frac{1}{2}=pq+\frac{p}{2}\Rightarrow 1=2pq+p\Rightarrow \frac{1}{p}=2q+1$
Hence, $k=1$