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Example 35 - Probability of a shooter hitting a target is 3/4 - Examples

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  1. Class 12
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Example 35 The probability of a shooter hitting a target is 3﷮4﷯ . How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99? Let X : Number of times he hits the target Hitting the target is a Bernoulli trial So, X has a binomial distribution P(X = x) = nCx 𝒒﷮𝒏−𝒙﷯ 𝒑﷮𝒙﷯ n = number of rounds fired p = Probability of hitting = 3﷮4﷯ q = 1 – p = 1 − 3﷮4﷯ = 1﷮4﷯ Hence, ⇒ P(X = x) = nCx 𝟑﷮𝟒﷯﷯﷮𝒙﷯ 𝟏﷮𝟒﷯﷯﷮𝒏−𝒙﷯ We need to find How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99 So, given P(X ≥ 1) > 99%, we need to find n Now, P(X ≥ 1) > 99 % 1 − P(X = 0) > 99 % 1 − nC0 3﷮4﷯﷯﷮0﷯ 1﷮4﷯﷯﷮𝑛﷯> 0.99 1 − 1﷮4﷯﷯﷮𝑛﷯ > 0.99 ` 1 − 0.99 > 1﷮4﷯﷯﷮𝑛﷯ 0.01 > 1﷮ 4﷮𝑛﷯﷯ 4﷮𝑛﷯ > 1﷮0.01﷯ 𝟒﷮𝒏﷯ > 𝟏𝟎𝟎 We know that 44 = 256 So, n ≥ 4 So, the minimum value of n is 4 So, he must fire atleast 4 times

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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