ProbabilityHard
Question
For the three events A, B and C, P (exactly one of the events A or S occurs) = P (exactly one of the events B or C occurs) = P (exactly one of the events C or A occurs) = p and O (all the three events occurs simultaneously) = p2, where 0 < p < 
.Then, the probability of at three events A, B and C occurring is
.Then, the probability of at three events A, B and C occurring is Options
A.
B.
C.
D.
Solution
We know that
P(exacly one of A of B occurs)
= P(A) + P(B) - 2P(A ∩ B)
∴ = P(A) + P(B) - 2P(A ∩ B) = P ......(i)
Similarly, P(B) + P(C) - 2P(B ∩ C) = P ......(ii)
and P(C) + P(A) - 2P(C ∩ C) = P ......(iii)
On adding Eqs. (i), (ii)and (iii), we get
2[P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A)] = 3p
⇒ P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A)]
......(iv)
It also given that P(A ∩ B ∩ C) = p2 ......(v)
∴ P(at least one of the events A, B, and C occur)
= P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)
[from Eqs. (iv) and (v)]
P(exacly one of A of B occurs)
= P(A) + P(B) - 2P(A ∩ B)
∴ = P(A) + P(B) - 2P(A ∩ B) = P ......(i)
Similarly, P(B) + P(C) - 2P(B ∩ C) = P ......(ii)
and P(C) + P(A) - 2P(C ∩ C) = P ......(iii)
On adding Eqs. (i), (ii)and (iii), we get
2[P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A)] = 3p
⇒ P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A)]
......(iv) It also given that P(A ∩ B ∩ C) = p2 ......(v)
∴ P(at least one of the events A, B, and C occur)
= P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)
[from Eqs. (iv) and (v)]Create a free account to view solution
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