Suppose I flip coins until either the first coin comes down head or I have flipped 10 coins, all of which came down tail. How many times do I flip coins in expectation?
Is there a closed form expression for the expected number of trials for arbitrary success probabilities and upper bound on the number of trials?
In the example, the distribution of the number of trials approaches a geometric distribution with mean 2 when I continue flipping coins longer and longer. For the example, the expected number of trials is $\frac{509}{256}$, which is, unsurprisingly, smaller than $2$.
Best Answer
If you flip a coin with success probability $p$ until the first success, you expect $1/p$ trials. The probability for not getting a success in $n$ trials is $(1-p)^n$. If you stop after $n$ trials, you reduce the expected number of trials by $1/p$ with probability $(1-p)^n$. Thus the expected number of trials becomes $(1-(1-p)^n)/p$.