How2Day: Probabilities With Dice

Knowing probability can give you the edge in dice games. Tsuneo Yamashita/Photodisc/Getty Images
Yahtzee, Dungeons and Dragons, craps and many other games that include a bit of luck, use dice to introduce some chance into the game dynamic. Along with the familiar six-sided die, four, eight, ten, twelve and twenty sided dice are also used. You could even think of a coin as a two-sided die, since it follows the same rules of probability. Whether you are designing a game or try to figure out the odds of the game that you are playing, you must be able to calculate probabilities with dice.

Calculating the probability of throwing a given number with a single die is straightforward: It is simply one divided by the number of sides on the die. The odds of throwing a three on a six-sided die is 1/6, about 16 percent, and the odds of throwing an 11 on a 20 sided die is 1/20, exactly five percent. The odds of throwing a set of numbers is the size of the set divided by the number of sides on the die. For example, the probability of throwing a five or a six on a six sided-die is 2/6, about 33 percent. The odds of throwing an even number is always 50%, no matter what die you use.

The odds that you will throw snake eyes--double ones on a pair of six-sided dice--you multiply the odds of throwing each one and multiply them. So the probability is (1/6 * 1/6), or about three percent. If you want to throw different numbers then you double the odds because the dice are interchangeable; it doesn't matter if you throw a one and a two or a two and a one. For example, the probability of getting a five and a six is (1/6 * 1/6 * 2), or about six percent.

For more than two dice, the situation becomes somewhat complicated. If you must get all the same number than simply multiply all the probabilities together. The chances of getting five ones on a six-sided dice is 1/6 raised to the fifth power--less than a hundredth of a percent. If you need all different numbers than you must multiply by the number of combinations available. For two dice there are two combinations, or permutations, with three dice there are six permutations, with four dice there are 24, and so forth. The number of permutations can be calculated using the factorial function, written as an exclamation mark (such as n!). The factorial function is the product of all whole numbers up to the number itself. For example, 6! = 1 * 2 * 3 * 4 * 5 * 6.

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