Say you have $n$ to $m$ odds of winning. The expected value is computed as
$$
E[profit] = p_{winning}*n + (1-p_{winning})*-m
$$
I am confused why it is $p_{winning}*n$ instead of $p_{winning}*(n+m)$. Doesn't the former imply that you don't get your $m$ back if you win?
Best Answer
No, because we're talking about the expected profit or loss, so the return of the stake doesn't enter into it.
Here's an example. Say a roulette wheel has $18$ black numbers, $18$ red numbers and $1$ green number. If you bet on black the probability of winning is $\frac{18}{37}$ and the house pays even money, that is $1$ to $1$. In $37$ plays, we expect to win $18$ times and lose $19$ times so, the expectation is $-\frac1{37}$. We have $n=m=1$, and the formula gives$$1\cdot\frac{18}{37}-1\cdot\frac{19}{37}=-\frac1{37}$$ as claimed.