Question

A game has an expected value to you of ​$100. it costs ​$500 to​ play, but if you win you receive​ $100,000 (including your ​$500 ​bet), for a net gain of ​$99 comma 500. what is the probability of​ winning?

Answer 1

Let $X$ be a random variable denoting the amount of winnings in any given round of the game. There is some probability $p$ of winning where $P(X=99,500)=p$, while $P(X=-500)=1-p$, assuming these are the only two possible outcomes. So we have

$\mathbb E[X]=\displaystyle\sum_xxP(X=x)=99,500p-500(1-p)=100$

$100,000p-500=100\implies p=\dfrac3{500}=0.006$