In 1654, Chevalier de Mere, a French gambler, wrote to Pierre Fermat and Blaise Pascal, two of France's mathematical giants, with a number of problems concerning the odds of particular combinations of numbers occurring, when several dice are rolled. This event is considered to be the birth of probability theory.

Let's investigate a simple question that Chevalier de Mere could have asked. Suppose we roll two dice. We can get a sum of 4 in two different combinations: (1,3) and (2,2). We can get a sum of 5 in two different combinations also: (1,4) and (2,3). Why is it that in de Mere's practice 5 appears more often than 4?

The answer is the following: the combinations (1,3) and (2,2) are not equiprobable. We have a probability of 1/6 that the first die rolls 2, and a probability of 1/6 that the second die rolls 2, thus making a combination (2,2) with the probability 1/36. By a similar argument we see that the probability that the first die rolls 1 and the second die rolls 3 is 1/36. The probability that the first die rolls 3 and the second die rolls 1 is also 1/36. Hence, the combination (1,3) is rolled with probability 2/36 = 1/18.

In the table below, the numbers in the left column show what is rolled on the first die and the numbers in the top row show what is rolled on the second die. We will color in blue the cells corresponding to the sum of 4, and in pink the cells corresponding to the sum of 5.

Probabilities for Two Dice

Now we can see that the sum 4 will be rolled with probability 3/36 = 1/12, and the sum 5 with probability 4/36 = 1/9.

Below you can check our random "roll of dice" generator. It will count for you the total number of rolls and the total for each sum. To set the count back to 0, press "Start Over" button.

Random Dice Generator