No heads in a random number of flips

Let $N \sim \text{Poisson}(2)$ , and flip a fair coin $N$ times. What is the probability that no heads appear? (Give a decimal.)

Show hints (2)+

Condition on $N$ : $\mathbb{P}[\text{no heads}]=\mathbb{E}[(\tfrac12)^N]$ .
Poisson PGF: $\mathbb{E}[z^N]=e^{\lambda(z-1)}$ with $z=\tfrac12,\lambda=2$ gives $e^{-1}$ .

Answer

Reveal answer →

0.3679 (± 0.002)

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