A gambler goes to the casino and plays a slot machine, he tells himself that he will only play 5 rounds, but if he wins the 5th round he keeps playing until he lose.
let $0< p < 1$ be the gambler's probabilty to win a single round, X is the number of rounds he played and Y the number of rounds he lost.
how can i calcuate E[X] and E[Y]?
My thoughts:
I know that the gambler will play at least 5 rounds in every situation,so i know there's Geometric Distribution from the 5th round, but i dont really know how to proceed.
Let there be another (virtual) gambler (with the same chance of winning in a round) who decides to play until his first loss has arrived and let $Z$ denote the number of rounds he plays.
Then we can say that $X$ has the same distribution as $5+UZ$ where $U=1$ if the original gambler wins the fifth round and $U=0$ otherwise.
Moreover $U$ and $Z$ are independent so that: $$\mathbb{E}X=5+\mathbb{E}U\mathbb{E}Z$$
Can you find $\mathbb{E}Z$ and $\mathbb{E}U$ yourself?
If $V$ denotes the number of losses of the original gambler in the first $4$ rounds then the total number of losses will be $1+V$ so that $$\mathbb EY=1+\mathbb EV$$ This because the first loss after the first $4$ rounds will make him end the game.
Can you find $\mathbb{E}V$ yourself?