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Question
One-shot is fired from each of the three guns. Let A, B, and C denote the events that the target is hit by the first, second and third guns respectively. assuming that A, B, and C are independent events and that P(A) = 0.5, P(B) = 0.6, and P(C) = 0.8, then find the probability that at least one hit is registered.
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Solution
A be the event that first gun hits the target
B be the event that second gun hits the target
C be the event that third gun hits the target
P(A) = 0.5, P(B) = 0.6, P(C) = 0.8
∴ P(A') = 1 − P(A) = 1 − 0.5 = 0.5
P(B') = 1 − P(B) = 1 − 0.6 = 0.4
P(C') = 1 − P(C) = 1 − 0.8 = 0.2
Now A, B, C are independent events
∴ A', B', C’ are also independent events.
∴ P (at least one hit is registered)
= 1 – P(no hit is registered)
= 1 – P(A' ∩ B' ∩ C')
= 1 – P(A') P(B') P(C')
= 1 – 0.5 × 0.4 × 0.2
= 1 – 0.04
= 0.96
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