How dice probabilities work

When you roll a standard die, each face has a 1 in 6 chance of coming up. With two dice, there are 36 possible combinations, but not all totals are equally likely: 7 is the most common because it can be rolled in 6 ways (6/36, or 1 in 6). The least likely totals are 2 and 12, with just 1 way to roll each.

One die

If the die is properly made, each face has the same probability: 1/6, or around 16.7%. Rolling a 4 three times in a row changes nothing for the next roll. We explain this in the gambler’s fallacy.

Two dice

Here’s where it gets interesting. With two dice, there are 6 × 6 = 36 outcomes. Look at how many ways there are to roll each total:

• 2 and 12: 1 way each.
• 3 and 11: 2 ways each.
• 4 and 10: 3 ways each.
• 5 and 9: 4 ways each.
• 6 and 8: 5 ways each.
• 7: 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).

That’s why so many board games revolve around 7.

What about doubles?

There are 6 ways out of 36 to roll doubles (two matching numbers): a 1 in 6 chance. In parchís (Spanish Ludo), rolling three doubles in a row happens roughly once every 216 attempts.

The same idea with cards

Cards have one important difference: there’s no replacement. If the as de oros (ace of coins) has already been drawn, it won’t appear again in that hand. That’s why remembering which cards have appeared is useful, whereas remembering previous dice rolls gives you no advantage.