A probability problem by A Former Brilliant Member

Probability Level 3

Marvin is a born sniper. Due to aging, he hits 80% of all his targets. His next mission is to shoot 8 persons. What is the probability (in percentage) that he will hit exactly 5 persons in his next mission?

Give your answer to the nearest integer.

16% 17% 15% 14%

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A Former Brilliant Member by A Former Brilliant Member

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2 solutions

His chance of hitting people is binomially distributed. So let X be the amount of people hit, then:

P(X=5)=8 nCr 5 x 0.8^5 x 0.2^3 = 0.147.

So the nearest percentage is 15%

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Joshua Lowrance
Apr 24, 2020

The probability of hitting 5 5 people out of 8 8 is ( . 8 ) 5 ( . 2 ) 3 (.8)^{5}(.2)^{3} ( 80 % 80\% chance he hits five of them and a 20 % 20\% chance he misses the other three).

However, there are ( 8 5 ) {8 \choose 5} ways of 5 5 out of 8 8 people getting hit.

Therefore, the answer is ( 8 5 ) ( . 8 ) 5 ( . 2 ) 3 = . 1468 ≃ 15 % {8 \choose 5}(.8)^{5}(.2)^{3}=.1468 \simeq 15\%

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