Probablity and Expected value

Question

Suppose you are playing a fair coin game and you win a dollar if a head shows up and lose a dollar if tail. what is the expected value of rounds you played before you lose the first dollar from your own pocket.

Answer 1

Note that one turn before you lose your first dollar, you must be even. Let $P(n)$ the probability of you being even after $n$ turns, then $$ P(n)=\begin{cases}0 & \text{if $n$ is odd}\\\begin{pmatrix}n\\\frac{n}{2}\end{pmatrix}\frac{1}{2^n} &\text{if $n$ is even} \end{cases}$$

Once you are even, there's $1/2$ chance you'll loose your first dollar. But we need to be careful though. Many of those combinations that get you even go through points were you already were below 0 before getting back even. For instance, you could have 4 turns such that you win, lose, lose, win (in that order).

Let $Q(n)$ be the probability that you are even for the first time. Then we can recursively construct it: $$Q(n) = P(n) \prod_{i=1}^{\frac{n}{2}-1}(1-Q(2i))\quad\text{with}\ Q(2)=\frac12$$

Since you are interested in finding the first time it loses from his pocket instead of the first time it crosses, then the probability you should work with is $R(n)=Q(n)/4$ to account that the first coin toss must be a win and the last coin toss must be a loss.

I will leave the simplification of these probabilities to you to compute the expected value.