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Anil can hit a balloon 6 times in 7 shots. Ben can hit the balloon 4 times in 5 shots, Carey can hit 3 times in 4 shots. What is the probability that the balloon is damaged by exactly 2 shots out of 3 shots?

Anil can hit a balloon 6 times in 7 shots. Ben can hit the balloon 4 times in 5 shots, Carey can hit 3 times in 4 shots. What is the probability that the balloon is damaged by exactly 2 shots out of 3 shots? Correct Answer 27/70

Given:

Anil can hit a balloon 6 times in 7 shots. Ben can hit the balloon 4 times in 5 shots, Carey can hit 3 times in 4 shots

Concept used:

The classical definition of probability.

Formula used:

Probability = Favorable Outcome / Total Outcome.

P(E) = n(E)/n(S)

Calculation:

We are supposed to find the probability of hitting the target by exactly two shots out of 3 shots.

So desired combinations are Anil, Ben, and Carey

Two shots hit the target in one of the following ways: 

(i) Anil and Ben hit and Carey fails to hit. 

(ii) Anil and Carey hit and Ben fails to hit. 

(iii) Ben and Carey hit and Anil fails to hit. 

The chance of hitting by Anil, P(A) = 6/7 and of not hitting by him P(A′) = 1 − (6/7) = 1/7​

The chance of hitting by Ben, P(B) = 4/5 and of not hitting by him P (B′) = 1 − (4/5) = 1/5​

The chance of hitting by Carey, P(C) = 3/4 and of not hitting by him P(C′) = 1 − (3/4) = 1/4

so the probability that the target will be in (Miss, hit, hit)

Condition after the shooting = (1/7) × (4/5) × (3/4) = 12/140.

Probability of condition (hit miss hit) = (6/7) × (1/5) × (3/4) = 18/140.

Probability of condition (hit hit miss) = (6/7) × (4/5) × (1/4) = 24/140.

The sum of the three probability

⇒ (12/140) + (18/140) + (24/140) = 54/140 = 27/70

∴ The probability that the balloon is damaged by exactly 2 shots is 27/70.

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