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

• A. $p(1+q)=1$
• B. $q(1+p)=1$
• C. $pq=1$
• D. $\frac{1}{p}+\frac{1}{q}=1$

Answer 1

The correct option is A $p(1+q)=1$

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