Consider n coins, each of which independently comes up heads with probability p. Suppose that n is
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(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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