Math With Dice – Part 2: Probability of Rolling Each Number

As a follow-up to my last post about dice, today we’re going to be looking at some more interesting properties of rolling several dice. Yesterday I calculated the probability of rolling a 6 on several dice at a time, but now I’m going to explain what happens when we find the outcome of multiple dice, all added together. If, for example, we roll 2 dice, they each show a number from 1 to 6. But if we add them together, there will be more of some numbers than others. A 2 is hard to get because there is only one way of getting it – each die has to roll a 1, so the chance is 1 in 36 between the 2 dice. But 5 is easier, because you can get 5 with 1 and 4, 4 and 1, 2 and 3, or 3 and 2. That means that the chance of rolling a 5 is 4 in 36, or 1 in 9. Consider this square.

A light gray square with smaller red and blue squares inside

The blue area represents the chance of rolling a 5 with 2 dice. The red area is the chance of rolling a 2. The gray is every other number that you could roll. Even though there is an equal chance of each die rolling from 1 to 6, there is an unequal chance of rolling each number (and you obviously can’t roll a 0 or a 1 with 2 dice). Now let’s take a look at these probabilities graphed. I wrote a program that takes a number of dice and a number of rolls, and rolls those dice over and over, adding up the total each time, then graphs how many times each number was rolled (from the lowest number on the top to the highest number on the bottom). This was the result when 3 dice were rolled about 3,000 times:

A bell curve created by graphing the number of times each number is rolled on 3 dice

As you can see, a bell curve forms, with the most common numbers (around 10, in this case) in the middle, and the least common numbers (such as 3 and 18 – 1 and 2 can’t occur) on the sides. The numbers in the middle have the most ways that they can be rolled, and the numbers on the sides have the least. If we add more dice, we can roll more numbers, as this graph with 10 dice, rolled 25,000 times shows:

A bell curve created by graphing the number of times each number is rolled on 3 dice

The more times the dice are rolled, the more even the graph becomes. If you want to try the program for yourself, here’s a link. When you run it, it will ask how many dice to roll, and how many times to roll them. Then it will graph the numbers and list the number of times each number was rolled underneath. You can also look at the code itself. By the way, this is also my 50th blog post, so yay.