Statistics of rolling dice

An interactive demonstration of the binomial behaviour of rolling dice

Maths Statistics Binomial

If you roll a fair, 6-sided die, there is an equal probability that the die will land on any given side. That probability is 1/6. This means that if you roll the die 600 times, each face would be expected to appear 100 times. You can simulate this experiment by ticking the "roll automatically" button above.

Now imagine you have two dice. The combined result from a 2-dice roll can range from 2 (1+1) to 12 (6+6). However, the probability of rolling a particular result is no longer equal. This is because there are multiple ways to obtain certain results. Let's use 7 as an example. There are 6 different ways: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1, whereas the result 2 can only be obtained in a single way, 1+1. This means you are 6 times more likely to achieve a 7 than you are to achieve a 2.

As the number of dice increases, the difference in probability between the most likely and least likely gets larger. The probability of rolling six sixes is 1 in 46,656! If you leave the experiment running for a while, you begin to see the bar chart take on a unmistakable shape - that of the binomial distribution.