Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win.
Find P(W < 0).
I am not sure how to calculate the probability for this. I know that P(W > 0) is 1/2.
Find E[W] How does it equal 0?
On
He only wins if he wins the first gamble. He breaks even if he loses the first and wins the second. Every other sequence of results has at least two losses, so $\bf {W} \lt 0$
On
If we assume that the expectation $E(W)$ exists, we can calculate it by conditioning on the result of the first toss. If it is a win, we have won $1$ dollar. If it is a loss, we are down $1$ dollar, and the game starts again, so our conditional expectation is $E(W)-1$. Thus $$E(W)=\frac{1}{2}+\frac{1}{2}(E(W)-1).$$ Solve. We get $E(W)=0$.
P(W > 0) is 0.5 because there is a 0.5 chance the player will win on their first gamble and stop with a net profit.
P(W = 0) is 0.25 because there is a 0.5 chance the player will lose on their first gamble, then also a 0.5 chance they will win on their second gamble and stop with a profit of 0.
All other combinations of gambles result in a net loss, so P(W < 0) = $1 - $P(W > 0)$ - $P(W = 0)
The expected value can be obtained by summing the products of each profit and the probability of that profit. In this case, you have $1 * 0.5 + 0 * 0.25 + (-1) * 0.125 + (-2) * 0.0625$, etc. So your E(W) is the sum $$\sum_{n=0}^{\infty}{\frac{1-n}{2^{n+1}}}=0$$