Chapter 4, Problem 4.21TE

Consider n coins, each of which independently comes up heads with probability p. Suppose that n is large and p is small, and let $\lambda =np$. 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 $P\left\{X=1\right\}$ is correct (in all cases, y is a Poisson random variable with parameter $\lambda$ )

a. Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter $P\left\{X=1\right\}\approx P\left\{Y=1\right\}=\lambda e^{-\lambda }$

b. Because the total number of heads that occur when all n coins are rolled is approximately a Poisson random variable with parameter $\lambda$, and because we stop only when this number is positive, $P\left\{X=1\right\}\approx P\left\{Y=1|Y>0\right\}=\frac{\lambda e^{-\lambda }}{1-e^{-\lambda }}$

c. Because at least one coin comes up heads, X will equal I 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 $\left(n-1\right)p\approx \lambda$, $P\left\{X=1\right\}\approx P\left\{Y=0\right\}=e^{-\lambda }$.