First you have to know what is this Mathematical Expectation

Mathematical Expectation

Mathematical expectation is the amount a bet will average winning or losing. It is an extremely important concept for the trader because it shows him how to evaluate most trading systems.

Using mathematical expectation is also the best way to analyze your system. Let's say you are betting a friend $ l, 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 1-to-1, and you're betting $1-to-$1. 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. 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 times: 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 getting 2-to-1 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 traders will trade only when they have the best of it; when they have the worst of it, they pass.What does it mean to have the 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-1, but you're getting 2-to-1 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-$1 on a 4-to-1 proposition. So the odds are in your favor, you have the best of it, and you should take 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.