2d6 probability

2d6 probability table: totals from 2 to 12

The probability of an event is its number of matching outcomes divided by 36. Count ordered pairs: (1, 6) and (6, 1) are different outcomes because the first and second die have different results. Percentages below are rounded to two decimal places.

• 2 — 1 way — 2.78%
• 3 — 2 ways — 5.56%
• 4 — 3 ways — 8.33%
• 5 — 4 ways — 11.11%
• 6 — 5 ways — 13.89%
• 7 — 6 ways — 16.67%
• 8 — 5 ways — 13.89%
• 9 — 4 ways — 11.11%
• 10 — 3 ways — 8.33%
• 11 — 2 ways — 5.56%
• 12 — 1 way — 2.78%

Probability of rolling doubles with two dice

Doubles means both dice show the same number. The six matching outcomes are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6).

Divide those 6 outcomes by all 36 possible outcomes: P(doubles) = 6/36 = 1/6 ≈ 16.67%. Alternatively, whatever the first die shows, the second die has one matching face out of six.

A specific double is different from any double. Snake eyes, or double 1, has just one matching outcome: P(double 1) = 1/36 ≈ 2.78%. An even total is not necessarily doubles: (2, 4) totals 6 but the dice do not match.

Probability of rolling a 7: a worked example

For an exact total of 7, list the matching pairs: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1). There are 6 ways, so P(total = 7) = 6/36 = 1/6 ≈ 16.67%.

For an exact total of 8, the matching pairs are (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2). There are 5 ways, so P(total = 8) = 5/36 ≈ 13.89%.

For ‘7 or 11’, add the counts for the two distinct totals: 6 ways to roll 7 plus 2 ways to roll 11. P(total = 7 or 11) = (6 + 2)/36 = 8/36 = 2/9 ≈ 22.22%.

At least probabilities: rolling 7, 8 or 10 and higher

‘At least’ includes the target total. For at least 7, add the ways for totals 7, 8, 9, 10, 11 and 12: 6 + 5 + 4 + 3 + 2 + 1 = 21. P(total ≥ 7) = 21/36 = 7/12 ≈ 58.33%.

For at least 8, count totals 8 through 12: 5 + 4 + 3 + 2 + 1 = 15. P(total ≥ 8) = 15/36 = 5/12 ≈ 41.67%. This is also the probability of rolling more than 7, which excludes 7 itself.

For at least 10, count totals 10, 11 and 12: 3 + 2 + 1 = 6. P(total ≥ 10) = 6/36 = 1/6 ≈ 16.67%.

You can also subtract the outcomes below the target from 36. Totals 2 through 6 have 1 + 2 + 3 + 4 + 5 = 15 outcomes, so P(total ≥ 7) = 1 − 15/36 = 21/36. Use exact fractions before rounding percentages.

Why the curve peaks at 7

Seven is the only total that can be made by every face: 1+6, 2+5, 3+4 and their mirrors. Totals further from 7 have fewer partitions, which produces the familiar triangular distribution rather than a flat one.

The average roll of 2d6 is 7 and the standard deviation is about 2.42. Totals 5 through 9 account for 24 of 36 outcomes, exactly two thirds. These are long-run probabilities, not a guarantee for any short sequence of rolls.