A student appears for tests I, II and Ill. The student· is successful if he passes either in test I and II or test I and III. The probabilities of the student passing in tests I, II, III are $p, q, \frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2}$ , then

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$p = q = 1$

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

$p = 1, q = 0$

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

answer is C.

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Detailed Solution

If $A, B, C$ are events of passing tests I, II, III respectively.

Then $\frac{1}{2} = P [(A \cap B) \cup (A \cap C)]$

$\begin{array}{l} = P (A \cap B) + P (A \cap C) - P (A \cap B \cap C) \\ = P (A) P (B) + P (A) P (C) - P (A) P (B) P (C) \\ = p q + \frac{p}{2} - \frac{p q}{2} \Rightarrow p (q + 1) = 1 \end{array}$

Only (c) satisfies the relation.

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