# A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are 34,12 and 58. Find the probability that the target is exactly hit by two of - Mathematics and Statistics

Sum

A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are 3/4, 1/2 and 5/8. Find the probability that the target is exactly hit by two of them

#### Solution

Let A ≡ the event that A hits the target

B ≡ the event that B hits the target

C ≡ the event that C hits the target.

It is given that,

P(A) = 3/4, P(B) = 1/2, P(C) = 5/8

P(A') = 1 –  P(A) = 1/4

Similarly, P(B') = 1/2, P(C) = 3/8

Now, A, B, C are independent events

∴ A', B', C' are independent events

Let F ≡ the event that target is hit by exactly two of them

∴ F = (A ∩ B ∩ C') ∪ (A ∩ B' ∩ C) ∪ (A' ∩ B ∩ C)

Events in the brackets are mutually exclusive.

Also A, B, C, A', B', C' are independent

∴ P(F) = P(A ∩ B ∩ C') + P(A ∩ B' ∩ C) + P(A' ∩ B ∩ C)

= P(A)·P(B)·P(C') + P(A)·P(B')·P(C) + P(A')·P(B)·P(C)

= 3/4*1/2*3/8 + 3/4*1/2*5/8 + 1/4*1/2*5/8

= 9/64 + 15/64 + 5/64

= 29/64.

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