Mathematics vs. Chance in Gambling

All games of chance deal with probability that one event will occur versus some other events. This probability is referred to as the odds. Thus, when one flips a coin, the probability of heads versus tails is expressed as the ratio 1:1 or 1-to-1 odds. This means that one out of every two times, in a long series of trials, the coin will show heads. Similarly, the probability of tails is exactly the same. The odds are therefore said to be even.

If we examine a die that is used in the game of craps, we will see that it has six faces, each bearing a different number from one to six. If the die is a perfect cube, perfectly balanced, then the probability of any one number coming up is one out of six trials. The odds against that event occurring are five out of six trials. Therefore, the odds of the event happening are 1:6 or 5-to-1 against.

If you were to run a gambling casino, which played this game of throwing one die, and you were to pay off the winners at 5-to-1, you wouldn't lose any money, but would go broke from paying the operating costs. What keeps the casinos open is that they always pay off at something less than the true odds. They keep on collecting this "house percentage" on every bet, and that is how they make their money.

Some examples follow:

In roulette, the odds against any particular number coming up are 37-1, since there are thirty-eight numbers on the American wheel (0,00, and 1-36). The casino pays this bet at 35-1. Therefore, in theory, and in the long run, every time a bet is made on a number, the house is collecting 2/38ths, or 5.26%!

In craps, the odds against rolling a seven are 30:6; that is, there are thirty-six possible combinations of dice, six of which equal a total of seven. Therefore, the odds are 30(non-sevens) to 6(sevens) against the player, or 5-to-1. If you bet on a seven being thrown, and seven comes up, the house will only pay you four times your bet instead of five times. Some casinos deliberately try to confuse the player by printing "five for one" on the craps layout. This means that they give you four of theirs plus one of yours for your bet of one. In this case the house percentage is 1/6 or 16.67%.

A similar situation exists with slot machines where the number of the various symbols of each kind on the machine limits the probability of the various kinds of payoffs.

If you bet at the racetrack, you know that the odds are totaled after all the bets are in. The track takes its percentage off the top and divides the rest, by totalizator machine, among the winners.