The Mathematics of Gambling

Why the Odds of Winning are Against You

When you have tried your hand at gambling in a casino and won a couple of times it can be incredibly easy to become hooked, but if you gamble often the chances of winning are very small. It all comes down to mathematics that the casinos have working in their favor.

Take a simple game with two die for example. When both die are rolled the player wins if the total of the two sets of dots is 2, 3, 4, 10, 11 or 12. The house i.e. casino wins if the total of the two sets of dots is 5, 6, 7, 8 or 9.

While at first glance it appears that the odds of the player of winning are greater because they have 6 possible totals as opposed to 5 for the house, but on further investigation you will realize that the odds are actually stacked in the house’s favor. It all comes down to simple mathematics i.e. the number of combinations that can be rolled.

The Trick of the Math

Now for the sake of our example say that one dice is blue and the other one is red. There is only one way that a total of 12 can be rolled i.e. 2 6’s, but to make a total of 7 there are six possible combinations 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, and 6 and 1. This means that if we roll a number of times the odds of rolling a total of 7 are much higher than the odds of rolling a total of 12. As the house wins when a total of 7 is rolled the odds are stacked in their favor.

example: Markov Chain


Matrix P2 must be wrong as pointed out by various friends here before. Explanation seems to be that you take $25 to gamble two times, first hand lose is 0.5 in probability, win once and then lose once is 0.5×0.5 = 0.25 in probability, win twice is 0.5×0.5 =0.25 in probability; adding all these up will be 1 in probability. So, no case is possible to carry $50 after playing twice and thus the 2nd row 3rd column should be “0” instead of “.5”.

In the table below you can see all the possible combinations and the ways they can occur. The number of dots on the red dice is represented in Row 1 and the number of dots on the blue dice is represented in Column 1. The black text indicates the totals of the two die when added together.

The squares highlighted in yellow represent wins by the player and the squares highlighted in green represent wins by the house. While there are a total of 36 different possible combinations the odds of the house winning are double that of the player, with the house having 24 possible combinations and the player only having 12. In any given roll of the die the player has only a 33 percent chance of winning while the house has a 66 percent chance of the die rolling in its favor.

Now let’s take another, simpler example with one dice that has 2 of its 6 faces painted yellow and the other 4 painted green. The yellow represents a win for the player and the green represents a win for the house. Obviously the opportunity for the house to win is far greater than that of the player. The odds of a green face showing when the dice is rolled, is much greater.

This simple mathematical principle is used in all casino games, stacking the odds of winning in the house’s favor. The game designers insure that the odds of the house winning are much greater than the player’s. The player will win occasionally but obviously will lose more often than they win.