The expected value of magnitude of winning and losing when playing a game Suppose we play a game at a casino. There is a \$5 stake and three possible outcomes: with probability $1/3$ you lose your stake, with with probability $1/3$ the bank returns your stake plus \$5, and with probability $1/3$ the bank simply returns your stake. Let $X$ denote your winning in one play. And you play 1000 times. What would you expect the magnitude of your win or loss to be approximately? * * * Is this question asking the same as to find the $Var(S)$ where $S=X_1+\cdots X_{1000}$, so the final answer should be $1000\cdot Var(X)$?

The key word here is probably "expect". With a large number of trials, you would win on average $\langle X \rangle$ dollars per trial and you have 1000 trials, so you should expect to win about 1000$\langle X \rangle$ dollars overall. Note that angle brackets denote expectation of the random variable.

Var$(S)$, on the other hand, would give you a measure of how close to this mean you might actually get.