Why Two Dice Don’t Give Every Total the Same Chance

Roll two ordinary six-sided dice and add the numbers. The possible totals run from 2 to 12.

There are eleven totals, but they do not each have a one-in-eleven chance. The difference comes from how many ways each total can happen.

Start with the individual rolls

Imagine one die is blue and the other is white.

A blue 1 and a white 6 make 7. A blue 6 and a white 1 also make 7. They produce the same total, but they are different outcomes.

With six possibilities for each die, there are 36 ordered outcomes. If the dice are fair and independent, those outcomes are equally likely.

Count the ways to make each total

2: one way

3: two ways

4: three ways

5: four ways

6: five ways

7: six ways

8: five ways

9: four ways

10: three ways

11: two ways

12: one way

The counts add to 36. That is a useful check that nothing has been left out or counted twice.

Seven has the most routes:

1 + 6

2 + 5

3 + 4

4 + 3

5 + 2

6 + 1

Its probability is 6 out of 36, or one in six. Twelve has only one route, 6 + 6, so its probability is one in 36.

Make three predictions

Before looking at the answers below, decide:

1. Which is more likely: a total of 7, or a total of at least 10?

2. Which is more likely: a total below 7, or a total above 7?

3. If ten rolls have produced no sevens, does the chance of a seven increase on the next roll?

Write down the reason for each answer. The reasoning is more useful than a lucky guess.

Try a 36-roll experiment

Make a list of the totals from 2 through 12. Roll two dice 36 times, adding one tally beside the total after each roll.

At the end, check that your tallies add to 36. Then compare the pattern with the outcome counts above.

You should not expect exactly six sevens and exactly one twelve. Those are expected counts, not a schedule the dice must follow. A fair experiment can produce an uneven result, especially with relatively few rolls.

If several people run the experiment, combine their counts and record the larger sample size. You will have more observations, although even a larger sample need not match the theoretical proportions exactly.

Answer key

1. Seven and at least ten are equally likely.

Ten has three routes, eleven has two and twelve has one. Together, that is six routes, the same number as seven. Each event has probability 6 out of 36.

2. Below seven and above seven are equally likely.

Below seven has 1 + 2 + 3 + 4 + 5 = 15 routes. Above seven has 5 + 4 + 3 + 2 + 1 = 15 routes.

The remaining six routes produce exactly seven.

3. No, assuming fair dice and independent rolls.

The next roll does not compensate for earlier ones. Its chance of producing seven remains one in six.

The useful habit

Before assigning probabilities, identify the outcomes that are actually equally likely.

Here, the ordered rolls are equally likely. The totals group those rolls into unequal-sized sets. Counting those sets explains the pattern without needing a simulation.

This illustration assumes fair six-sided dice and independent rolls. Loaded dice or dependent outcomes require a different calculation.

The same care with what you count matters outside probability. Before You Compare Two Numbers shows why counts, percentages and percentage-point changes can tell different stories.

The outcome counts and answer calculations are included so readers can check the reasoning.