Question

A student appears for tests $I,II$ and $III$. The student is successful if he passes in tests $I,II$ or $I,III$. The probabilities of the student passing in tests $I,II$ and $III$ are respectively $p,q$ and $\frac{1}{2}$ . If the probability of the student to be successful is $\frac{1}{2}$, then find the relationship between $p$ and $q$ ? Also find the value of $q$ when $p=\frac{2}{3}$.

Answer 1

Given, $P\left(I\right)=p$
$P\left(II\right)=q$
$P\left(III\right)=\frac{1}{2}$

According to the question student have to pass the test $I$ and any one of $II,III$ or both of them
$\therefore P\left(I\right)P\left(II\right)P\left(II{I}^{\prime }\right)+P\left(I\right)P\left(I{I}^{\prime }\right)P\left(III\right)+P\left(I\right)P\left(II\right)P\left(III\right)=\frac{1}{2}$
$\Rightarrow p.q(1-\frac{1}{2})+p.(1-q).\frac{1}{2}+p.q.\frac{1}{2}=\frac{1}{2}$
$\Rightarrow pq+p(1-q)+pq=1$
$\Rightarrow 2pq+p-pq=1$
$\Rightarrow pq+p=1$
$\Rightarrow p(q+1)=1$

Now, when $p=\frac{2}{3}$
$\Rightarrow \frac{2}{3}(q+1)=1$
$\Rightarrow 2q+2=3$
$\Rightarrow q=\frac{1}{2}$