Expectation and Hourly Rate - Mathematical Expectation

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 lime it comes up tails, you lose. The odds of its coming up heads are 1-to-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.
If you can manage 500 flips of the coin per hour, your hourly rate is now $250, because on average you will lose one dollar 250 times and win two dollars 250 times $500 minus $250 equals a $250 net win. Notice again that your mathematical expectation, which is the amount you will average winning per bet, is 50 cents. You have won $250 after betting a dollar 500 limes: That works out to be 50 cents per bet.
Mathematical expectation has nothing to do with results. The imbecile might win the first ten coin flips in a row, but gelling 2-to-l odds on an even-money proposition, you still earn 50 cents per $1 bet. It makes no difference whether you win or lose a specific bet or series of bets as long as you have a bankroll to cover your losses easily. If you continue to make these bets, you will win, and in the long run your win will approach specifically the sum of your expectations.
Anytime you make a bet with the best of it, where the odds are in your favor, you have earned something on that bet, whether you actually win or lose the bet. By the same token, when you make a bet with the worst of it, where the odds are not in your favor, you have lost something, whether you actually win or lose the bet.
You have the best of it when you have a positive expectation, and you have a positive expectation when the odds are in your favor. You have the worst of it when you have a negative expectation, and you have a negative expectation when the odds are against you. Serious gamblers bet only when they have the best of it; when they have the worst of it, they pass.
What does it mean to have me odds in your favor? It means winning more on a result than the true odds warrant. The true odds of a coin's coming up heads are 1-to-l, but you're getting 2-to-l for your money. The odds in this instance are in your favor. You have the best of it with a positive expectation of 50 cents per bet.
Here is a slightly more complicated example of mathematical expectation. A person writes down a number from one to five and bets $5 against your $1 that you cannot guess the number. Should you take the bet? What is your mathematical expectation?
Four guesses will be wrong, and one will be right, on average. Therefore, the odds against your guessing correctly are 4-to-1. Chances are that in a single try you will lose the dollar. However, you are getting $5-to-$l on a 4-to-l proposition. So the odds are in your favor, you have the best of it, and you should lake the bet. If you make that bet five times, on average you will lose $ 1 four times and win $5 once. You have earned $1 on five bets for a positive expectation of 20 cents per bet.
A bettor is taking the odds when he stands to win more than he bets, as in the example above. He is laying the odds when he stands to win less than he bets. A bettor may have either a positive or a negative expectation, whether he is taking the odds or laying them. If you lay 550 to win $10 when you are only a 4-to-1 favorite, you have a negative expectation of $2 per bet, since you'll win $10 four times but lose $50 once, on average, for a net loss of $10 after five bets. On the other hand, if you lay $30 to win $10 when you're a 4-to-1 favorite, you have a positive expectation of $2, since you'll win $10 four times again but lose only $30 once, for a net profit of $10. Expectation shows that the first bet is a bad one and the second bet is a good one.
Mathematical expectation is at the heart of every gambling situation, When a bookmaker requires football bettors to lay $11 to win $10, he has a positive expectation of 50 cents per $10 bet. When a casino pays even money on the pass line at the craps table, it has a positive expectation of about $1.40 per $100 bet since the game is structured so that the pass line bettor will lose 50.7 percent of the time and win 49.3 percent of the time, on average. Indeed it is this seemingly minuscule positive expectation that provides casinos around the world with all their enormous profits. As Vegas World casino owner Bob Stupak has said, "Having one-thousandth of one percent the worst of it, if he plays long enough, that one-thousandth of one percent will bust the richest man in the world."
In most gambling situations like casino craps and roulette, the odds on any given bet are constant. In others they change, and mathematical expectation can show you how to evaluate a particular situation. In blackjack, for instance, to determine the right play, mathematicians have calculated your expectation playing a hand one way and your expectation playing it another way. Whichever play gives you a higher positive expectation or a lower negative expectation is the right one For example, when you have a 16 against the dealer's 10, you're a favorite to lose. However, when that 16 is 8,8, your best play is to split the 8s, doubling your bet. By splitting the 8s against the dealer's 10, you still stand to lose more money than you win, but you have a lower negative expectation than if you simply hit every time you had an 8,8 against a 10.