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. $p=1,q=\frac{1}{2}$

Answer 1

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

Let A, B and C denote the events of passing the tests I, II and III, respectively.
Evidently A, B and C are independent events.
According to given condition,
$\frac{1}{2}=P\left[\right(A\cap B)\cup (A\cap C\left)\right]=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)=P\left(A\right).P\left(B\right)+P\left(A\right).P\left(C\right)-P\left(A\right).P\left(B\right).P\left(C\right)=pq+p.\frac{1}{2}-pq.\frac{1}{2}\Rightarrow 1=2pq+p-pq\Rightarrow 1=p(q+1)$ ...(i)
The values of option (c) satisfy Eq. (i).
[Infact, Eq.(i) is satisfied for infinite number of values of p and q. 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, $0<\frac{1}{q+1}\le 1i.e.0<p\le 1.$ But we have to choose correct answer from given ones.]