Mathematical expectation is the amount a bet will average winning or losing. It is an extremely important concept for the gambler because it shows him how to evaluate most gambling problems. Using mathematical expectation is also the best way to analyze most poker plays. Let’s say you are betting a friend $ 1, even money, on the flip of a coin. Each time it comes up heads, you win; each time it comes up tails, you lose.
The odds of its coming up heads are 1to-1, and you’re betting $l-to-$l. Therefore, your mathematical expectation is precisely zero since you cannot expect, mathematically, to be either ahead or behind after two flips or after 200 flips. Your hourly rate is also zero. Hourly rate is the amount of money you expect to win per hour. You might be able to flip a coin 500 times an hour, but since you are getting neither good nor bad odds, you will neither earn nor lose money. From a serious gambler’s point of view, this betting proposition is not a bad one.
It’s just a waste of time. But let’s say some imbecile is willing to bet $2 to your $1 on the flip of the coin. Suddenly you have a positive expectation of 50 cents per bet. Why 50 cents? On the average you will win one bet for every bet you lose. You wager your first dollar and lose $1; you wager your second and win $2. You have wagered $1 twice, and you are $1 ahead. Each of these $1 bets has earned 50 cents.