Question

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.

Answer 1

`P(A) = 4/5; P(B) = 3/4; P(C) = 2/3`

`P(A') = 1/5; P(B') = 1/4; P(C') = 1/3 `

1) P (Exactly two person hit the target)

= 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)

`= 4/5 xx 3/4 xx 1/3 + 4/5 xx 1/4 xx 2/3 + 1/5 xx 3/4  xx 2/3`

`= 12/60 + 8/60 + 6/60`

`= 26/60 = 13/30`

2) P (At least two person hit the target)

= P (Two person hit the target) + P (All three hit the target)

`= 26/60 + 4/5 xx 3/4 xx 2/3`

`= 26/60 + 24/60`

`= 50/60 = 5/6`

3) P (None hit the target)

= P(A' ∩ B' ∩ C')

`= P(A'). P(B').P(C')`

`= 1/5 xx 1/4 xx 1/3 = 1/60`