Gambling Mathematics: Why Will The Casino Win in the Long Run

Gambling in a casino and winning once or twice might hook you, but if you play often, there is a very small chance that you would win. Why? Because the games are mathematically designed for the casino to win.

To make it simple, let us use a simple die game for our discussion.  Two dice are rolled. If the sums of the number of dots are 2, 3, 4, 10, 11, and 12, the player wins. If the sums are 5, 6, 7, 8, and 9, the house (the casino) wins.

At first glance, the design seems to favor the player. There are 6 possible sums in favor of him, while there are only 5 possible sums in favor of the house. But looking closely, the design actually favors the house. The secret lies in the mathematics: the number of ways in getting the sums.

The Math Trick

Suppose the dice are colored red and blue. There is only one way to get a sum of 12 (6 + 6), while there are six ways to get a sum of 7: (1 + 6), (2 + 5), (3 + 4), (4 + 3), (5 + 2), (6 + 1). That means we are certain to have more 7’s than 12’s if we roll the two dice a large number of times.

The table below shows all the possible sums and how they can be obtained. The first row represents the number of dots on the red die, and the first column represents the number of dots on the blue die.  The  numbers in black texts represent the sums when the number of dots on the two dice are added.

Notice that the yellow squares are wins of players, while the green squares represent wins of the house. There are 36 possible sums — only 12 ways for the players, but 24 ways for the house. You see the difference? In a single roll, the probability that the player winning is 12/36 , and the probability that the casino winning is 24/36.


The design of the game above is equivalent to rolling a painted die — two faces yellow and four faces green (a die has 6 faces), and ignoring the dots. If the top face turns out to be green, then the casino wins; if yellow, then the player wins. Since there are more green faces than yellow faces, it is likely that  a green face will be rolled most of the time.

The principle used in designing the game above is the same principle used in  designing casino games. The designers of the games rely on theoretical probability to ensure the casino winning. That means that even if players win occasionally, they will certainly lose most of the time.

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