# Question

Consider n coins, each of which independently comes up heads with probability p. Suppose that n is large and p is small, and let λ = np. Suppose that all n coins are tossed; if at least one comes up heads, the experiment ends; if not, we again toss all n coins, and so on. That is, we stop the first time that at least one of the n coins come up heads. Let X denote the total number of heads that appear. Which of the following reasonings concerned with approximating P{X = 1} is correct (in all cases, Y is a Poisson random variable with parameter λ)?

(a) Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter λ,

P{X = 1} ≈ P{Y = 1} = λe−λ

(b) Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter λ, and because we stop only when this number is positive,

P{X = 1} ≈ P{Y = 1|Y > 0} = λe−λ / 1 − e−λ

(c) Because at least one coin comes up heads, X will equal 1 if none of the other n − 1 coins come up heads. Because the number of heads resulting from these n − 1 coins is approximately Poisson with mean (n − 1)p ≈ λ,

P{X = 1} ≈ P{Y = 0} = e−λ

(a) Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter λ,

P{X = 1} ≈ P{Y = 1} = λe−λ

(b) Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter λ, and because we stop only when this number is positive,

P{X = 1} ≈ P{Y = 1|Y > 0} = λe−λ / 1 − e−λ

(c) Because at least one coin comes up heads, X will equal 1 if none of the other n − 1 coins come up heads. Because the number of heads resulting from these n − 1 coins is approximately Poisson with mean (n − 1)p ≈ λ,

P{X = 1} ≈ P{Y = 0} = e−λ

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