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 $p,q$ and $\frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2},$ then

• A. $p=q=1$
• B. $p=q=\frac{1}{2}$
• C. $p=1,q=0$
• D. $p=1,q=\frac{1}{2}$
• E. None of these

Answer 1

The correct option is C $p=1,q=0$

Let $A,B,C$ denote the events of passing the tests I, II and III respectively.
Evidently $A,B,C$ are independent $\frac{1}{2}=P[(A\cap B)\cup (A\cap C)]$
$=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)+P\left(A\right)P\left(C\right)-P\left(A\right)P\left(B\right)P\left(C\right)$
$=Pq+p.\frac{1}{2}-pq.\frac{1}{2}$ or $1=2pq+p-pq=p(q+1)$.
Of tbe given values, values in (C) satisfy (1).
Hence (c) is tbe required answer. [In fact, the equality (1) is satisfied for infinite no. of values of $p$ and $q$.
For, if we take any value of $q$ such that $0\le q\le 1,$ then $p$ takes the value $\frac{1}{(q+1)}$.
It is evident that $0<1/(q+1)\le 1i.e.0<p\le 1.$
But we have to correct the answer from given ones.