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

There are infinitely many values of p and q

Let A, B and C be the events that the student is successful in test I, II and III respectively, then P (the student is successful)

$=P[(A ∩ B ∩ C') ∪ (A ∩ B' ∩ C) ∪ (A ∩ B ∩ C)= P(A ∩ B ∩ C')+ P(A ∩ B' ∩ C)+P(A ∩ B ∩ C)$

$= P(A).P(B).P(C)+P(A).P(B').P(C)+P(A).P(B).P(C)$       [∵ A, B, C are independent]

$=pq \left(1-\frac{1}{2}\right) +p(1-q)\left(\frac{1}{2}\right)+pq\left(\frac{1}{2}\right)=\frac{1}{2}p(1+q)⇒ \frac{1}{2}=\frac{1}{2}p(1+q)⇒  p(1+q)=1.$

This equation has infinitely many values of p and q.