91Ó°ÊÓ

At time $0$, a coin that comes up heads with probability $p$ is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate $\mathrm{λ}$, the coin is picked up and flipped. (Between these times, the coin remains on the ground.)

Since we select our times of flipping a coin according to the Poisson process with the rate $\mathrm{λ}$, the number of times that we flip a coin before has distribution Pois $\left(\mathrm{λ}t\right)$. Name that random variable $Y$. If $Y=0$, by the definition, the probability that we will finish with Head on is 1 since we do not flip a coin and we are told that we begin with Head. If $Yâ‰\yen 1$, the probability that we finish with Heads is simply $p$since it only depends on the final throw. Hence, the probability is

$p=P($Head$âˆ\pounds Y=0\left)P\right(Y=0)+P($Head $âˆ\pounds Yâ‰\yen 1\left)P\right(Yâ‰\yen 1)$

$=e^{-\mathrm{λ}t}+p·\left(1-e^{-\mathrm{λ}t}\right)$

The required probability is$e^{-\mathrm{λ}t}+p·\left(1-e^{-\mathrm{λ}t}\right)$