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A student takes tests I, II and III. He is said to be successful if he clears either both tests I and II, or both tests I and III. The probabilities of passing tests I, II and III are p, q and 1/2 respectively. If the probability that the student is successful is 1/2, then the relation between p and q is

1. pq + p = 1
2. p² + q = 1
3. pq - 1 = p
4. none of these

Correct answer: pq + p = 1

Solution

With independence, P(success)=pq + p(1/2) - pq(1/2) = (p/2)(1+q). Setting this to 1/2 gives p(1+q)=1, i.e. pq + p = 1.

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