Question

A students appears for tests $I,II$ and $III$. The students is successful if he passes either in tests $I$ and $II$ or tests $I$ and $III$. The probabilities of the students passing in test $I,II,III$ are $p,q$ and $\frac{1}{2}$ respectively. The probability that the student is successful is $\frac{1}{2}$ then the relation between $p$ and $q$ is given by

• A. $pq+p=1$
• B. $p^2+q=1$
• C. $pq-1=0$
• D. None of these

Answer 1

The correct option is A $pq+p=1$

Let $A,B$ and $C$ be the vent that the students is successful in test $I,II$ and $III$ respectively. Then
$P($the student is successful$)$
$=P\left(A\right)P\left(B\right)(1-P\left(C\right))+P\left(A\right)(1-P\left(B\right))P\left(C\right)+P\left(A\right)P\left(B\right)P\left(C\right)$
$=p.q(1-\frac{1}{2})+p(1-q)\frac{1}{2}+p.q.\frac{1}{2}=\frac{1}{2}pq+\frac{1}{2}p(1-q)+\frac{1}{2}pq$
$=\frac{1}{2}(pq+p-pq+pq)=\frac{1}{2}(pq+p)$
$\therefore \frac{1}{2}=\frac{1}{2}(pq+p)\Rightarrow 1=pq+p$