Dice Roll

The Problem

A die has six sides, numbered one through six. When you roll the die, each side has an equal chance of coming up. When you roll two dice, the total shown can be anywhere from two to twelve. When you roll three dice, the totals can range from three to eighteen. What are the chances of each total occurring?

The Analysis

Assuming the die is not tainted, it has an equal chance of coming up on any side. Therefore, a single die should display a given number one of every six times or about 16.67%. For two dice, there are 36 (6 x 6) different ways they can come up. Here are just four examples:

Die ADie B

The only way to get a total of 2 is for both dice to display 1. Similarly, the only way to get a total of 12 is for both dice to display 6. Therefore, each of these has just one chance in 36 (about 2.78%) of coming up in a roll. However, other totals can be formed in multiple ways. For example, as show in the above table, 3 can be formed in two ways (1 + 2, or 2 + 1). Therefore, the chance of two dice totaling 3 is 2 in 36, or about 5.56%. There are six different ways to form the total 7 (1 + 6, 2 + 5, etc.). Therefore the chance of 7 coming up from two dice is 6 in 36 or about 16.67%.

With three dice, there are 216 (6 x 6 x 6) different ways the dice can come up. The most likely rolls are 10 and 11.

See the Expected column within the game for the odds on each total.

Playing the Game

You can choose to roll two or three dice or only one. An image of one, two, or three dice appears near the top left corner of the window. To roll the dice, click on the image. To roll again, click on it again. The actual and expected results are shown in the bottom half of the window. You can choose to see just results for the current session or to see historical results. (The historical results for one and two dice are since about June 1, 2003. The results for three dice are since about August 1, 2007.) The results are updated after each roll.

To automatically roll the dice multiple times, enter a number in the field to the right of the dice and then click the Auto Roll button. The program automatically performs the number of rolls you specified and updates the displayed results. If you enter an especially high number, you might experience some delay as the rolls are processed.

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