Probability of Binomial twice of Geometric
I've come up with an interesting result:
Let X be the amount of failures of Bernoulli(p) until we get (p).
I found it using the Taylor expansion of , where the coefficient of turns out to be .
I would like to see a probabilistic proof of this result.
Explanation of the process in words:
Roll a die with probability p of getting "X". Each time that we don't get "X", toss 2 balanced coins and accumulate the number of heads and tails. When you get "X", check if you got the same amount of heads and tails.
If we succeed immediately (with probability p), , and , thus , thus contributing p to the conditional sum, , and everything is alright.