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Question

An archer shoots arrows at a circular target where the central portion of the target inside is called the bull.The archer hits the bull with probability 1/32.Assume that the archer shoots 96 arrows at the target,and that all shoots are independent.Find the pmf of the number of bulls that the archer hit.Give an approximation for the probability of the archer hitting no more than one bull.

Answer 1

This appears to be a binomial random variable in that there are multiple independent trials with equal probability. So the pmf would be a binomial pmf. These follow the following format:

P(X = x) = (_nC_x) p^x • (1 - p)^{n - x}

So in this case, it would look like this:

P(X = x) = (₉₆C_x) (^1/₃₂)^x • (³¹/₃₂) ^{96 - x}

For the second part of the problem, we're looking for the probability of no more than one success, or P(X ≤ 1).

This is the sum of P(X = 0) and P(X = 1).

These are computed as follows:

(₉₆C₀) (^1/₃₂)⁰ • (³¹/₃₂) ⁹⁶ + (₉₆C₁) (^1/₃₂)¹ • (³¹/₃₂) ⁹⁵ = (³¹/₃₂) ⁹⁶ + (96)(^1/₃₂) • (³¹/₃₂) ⁹⁵ ≈ 0.4746 + 0.1470 ≈ 0.1944.