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 th at 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

Answer: (C)

p = 1, q = 0

Answer 2

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,
$\frac{1}{2}=P[(A \cap B)\cup (A \cap B)]$
$=P(A \cap B)+P(A \cap C)-P(A \cap B \cap C)$
= P {A) P(B) + P(A) . P(C) - P(A) . P(B) . P(C)
=pq+p $\frac{1}{2}-p\frac{1}{2}$
$\Rightarrow \,$1=2 p q + p - p q $\Rightarrow \,$ 1 = p(q + 1) $\, \, \, \, \, \, \, \, \, \, \, \,$ ....(i)
The values of option (c) satisfy E (i).
[ Infact, E (i) is satisfied for infinite number of values of p and If we take any values of q such that $0 \le q \le 1.$
then, p takes the value $\frac{1}{q+1}$ It is evident that,
$0 < \frac{1}{q+1} \le 1 \, i.e., 0 < p \le 1$ But we have to choose correct answer from given ones.]