Poisson processes and coin flips?

Question

> At time 0, a coin that lands on heads with probability p is flipped and lands on heads. At times chosen with a Poisson process of rate lambda, the coin is flipped again. What is the probability that the coin is on its head at time t? Note: Flip means I toss the coin again. I'm having trouble with parsing this question, but also I don't know how I'd solve it any way I think about it. Am I renewing the Poisson process every time I flip? As in, flip, generate Poisson random number, flip again, generate Poison random number, flip again, etc. Or is the number generated by the Poisson process the regular interval between flips? And either way, how would I solve it?

Answer 1

It seems like a bit of a trick question.

The key is to identify the set of possible states of the coin:

{Heads, Tails, Neither}

Neither is the tricky one, and here means "Coin is being flipped, i.e. in the air"

p and λ are sufficient to answer the question