ProbabilityHard
Question
A student appears for tests I, III. The student is successful if he passes either in tests I and III. The probabilities of the student passing in tests I, II and III are p, q and 
respectively . If the probability that the student is successful, is , then
respectively . If the probability that the student is successful, is , thenOptions
A.p = q = 1
B.p = q =
C.p = 1, q = 0
D.p = 1, q =
Solution
Let A, B, C denote the events of passing the tests I, II and III respectively.
Evidently A, B, C are independent events.
According to given condition
= P[(A ∩ B) ∪ (A ∩ C)]
= P(A ∩ B) + (A ∩ C) - P (A ∩ B ∩ C)
= P(A)P(B) + P(A).P(C) - P(A).P(B).P(C)
= pq + p.
- pq.
⇒ 1 = 2 pq + p - pq
⇒ 1 = p(q + 1)
The values of option (c) satisfy Eq. (i).
Hence, (c) is required answer
{In fact, equality (i) is satisfied for infinite num,ber of values of p and q For if ew take any values of q such that 0 ≤ 0 ≤ 1then p takes the value
It is evident that 0 < ≤ 1 ie.,
0 ≤ p ≤ 1. But ew have to cheose correct answer from give ones.}
Evidently A, B, C are independent events.
According to given condition
= P[(A ∩ B) ∪ (A ∩ C)]= P(A ∩ B) + (A ∩ C) - P (A ∩ B ∩ C)
= P(A)P(B) + P(A).P(C) - P(A).P(B).P(C)
= pq + p.
- pq. ⇒ 1 = 2 pq + p - pq
⇒ 1 = p(q + 1)
The values of option (c) satisfy Eq. (i).
Hence, (c) is required answer
{In fact, equality (i) is satisfied for infinite num,ber of values of p and q For if ew take any values of q such that 0 ≤ 0 ≤ 1then p takes the value
It is evident that 0 < ≤ 1 ie.,0 ≤ p ≤ 1. But ew have to cheose correct answer from give ones.}
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