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Question

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

A
p=1,q=0
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B
p=2/3,q=1/2
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C
p=3/5,q=2/3
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D
there are inifinitely many value of p and q
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Solution

The correct options are
A p=1,q=0
B p=2/3,q=1/2
C p=3/5,q=2/3
D there are inifinitely many value of p and q
Let A,B and C be the events that the student is successful in tests I, II and III, respectively. Then P (the student is successful)
=P[(ABC)(ABC)(ABC)]
=P(ABC)+(ABC)+(ABC)
=P(A)P(B)P(c)+P(A)P(B)P(C)+P(A)P(B)P(C)
[ A,B and C are independent]
=pq(11/2)+p(1q)(1/2)+(pq)(1/2)
=12[pq+p(1q)+pq]=12p(1+q)
12=12p(1+q)p(1+q)=1
This equation is satisfied for all pairs of values in (a), (b) and (c). Also, it is satisfied for inifinitely many values of p and q. For instance, when p=n/(n+1) and q=1/n, where n is any positive integer.

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