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 and III are, respectively, $p$, $q$ and $1/2$. if the probability that the student is successful is $1/2$, then $p(1+q)=$

• A. $\frac{1}{2}$
• B. $1$
• C. $\frac{3}{2}$
• D. $\frac{3}{4}$

Answer 1

The correct option is B $1$

If $A,B,C$ represent events that the student is successful in tests I, II, III, respectively, then the probability that the student is successful is
$P\left[\right(A\cap B\cap C^{\prime })\cup (A\cap B^{\prime }\cap C)\cup (A\cap B\cap C\left)\right]$
$=P(A\cap B\cap C^{\prime })+P(A\cap B^{\prime }\cap C)+P(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)P\left(C^{\prime }\right)+P\left(A\right)P\left(B^{\prime }\right)P\left(C\right)+P\left(A\right)P\left(B\right)P\left(C\right)$
$[\because A,B,C\text{are independent events}]$
$=pq(1-\frac{1}{2})+p(1-q)\frac{1}{2}+pq\frac{1}{2}$
$=pq-\frac{1}{2}pq+\frac{1}{2}p-\frac{1}{2}pq+\frac{1}{2}pq$
$=\frac{1}{2}(pq+p)$
$\therefore \frac{1}{2}(pq+p)=\frac{1}{2}$
$\text{or}p(1+q)=1$