Probability 3 Question 18

18. 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 12, respectively. If the probability that the student is successful, is 12, then

(a) p=q=1

(b) p=q=12

(c) p=1,q=0

(d) p=1,q=12

(1986, 2M)

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Solution:

  1. 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,

12=P[(AB)(AC)]=P(AB)+P(AC)P(ABC)=P(A)P(B)+P(A)P(C)P(A)P(B)P(C)=pq+p12pq121=2pq+ppq1=p(q+1)

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 values of q such that 0q1, then, p takes the value 1q+1. It is evident that, 0<1q+11 i.e. 0<p1. But we have to choose correct answer from given ones.]



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