Probability of Rolling Two Dice

Roll two dice and add them up. Which total is most likely? Let's find out with a picture. Here are two ordinary dice. Each one shows a number from one to six, and each number is equally likely.

Let's list every possible pair. Die one goes down the side, die two goes across the top. Six rows times six columns gives thirty six boxes. Each box is one equally likely outcome.

So there are thirty six equally likely outcomes in total. Remember that number. Now let's count the boxes that make a total of seven. Watch the diagonal light up. One and six, two and five, three and four, and the three reversed ones. That is six boxes.

• 36 outcomes in total
• Total 7: 6 ways

Probability means the favourable outcomes divided by all outcomes. Let's write that as a formula.

Favourable is the six boxes we counted. Total is the thirty six boxes on the whole grid. Now we put the numbers in for a total of seven.

Six over thirty six can be simplified. Divide the top and bottom by six.

That leaves one over six. So you get a seven about one time in every six rolls.

Next example: a total of four. Which pairs work? One and three, two and two, three and one.

Three over thirty six simplifies by dividing both by three.

That gives one over twelve. A four is less likely than a seven.

Why is seven the winner? Its diagonal is the longest one, so it has the most pairs. Totals near two or twelve have short diagonals. Only one box makes two, and only one makes twelve. Common mistake: thinking each total is equally likely. Eleven totals exist, but they do not share the thirty six boxes equally.

• 2 and 12: only 1 way
• Totals are NOT equally likely

Real life: in board games, a seven comes up most often, so players expect it.

Try one yourself. What is the probability of rolling a total of ten with two dice? To sum up: list all thirty six outcomes, count the favourable ones, and divide. Seven is the most likely total.