Dice and Coin Probability: Odds for 2d6, Doubles and Streaks

The one rule behind every table

A fair die has six equally likely faces, a fair coin two. Roll two dice and there are 6 × 6 = 36 equally likely combinations; flip five coins and there are 2⁵ = 32. The probability of an event is the number of combinations that produce it divided by the total. Everything below is this division, written out. When two events have to happen together and do not influence each other, multiply their probabilities; when you want the chance that something happens at least once, it is often easier to compute the chance that it never happens and subtract from 1.

Two dice: every total from 2 to 12

The 36 combinations are not spread evenly over the eleven totals. Only one combination makes a 2 (1+1) and only one makes a 12 (6+6), but six make a 7. Here is the full distribution; the last column is the chance of rolling that total or higher, which is what you need when a game asks for "8 or more".

Three facts fall out of the table. Seven is the single most likely total at 16.7%. The middle totals 6, 7 and 8 together cover 44.4% of all rolls. And the extremes are rare: 2 and 12 each come up once in 36 rolls on average. The dice roller accepts 2d6 and any other notation and keeps a history of your rolls, so you can watch the pile form around seven.

Doubles and specific pairs

There are six doubles (1+1 to 6+6) among the 36 combinations, so doubles come up with probability 6/36 = 1 in 6 on any roll. A specific double such as double sixes is 1/36 = 2.8%. A specific pair of different numbers, say a 3 and a 5, can happen two ways (3+5 and 5+3), so it is 2/36 = 5.6%, twice as likely as any double. The probability of rolling doubles three times running is (1/6)³ = 1/216 = 0.46%.

Three dice: totals from 3 to 18

With three six-sided dice there are 216 combinations and the curve is taller and narrower. Totals of 10 and 11 are the most likely at 12.5% each; 3 and 18 are 1 in 216.

At least one six

The chance of no six on one die is 5/6. On n dice, the chance of no six at all is (5/6)ⁿ, so the chance of at least one six is 1 − (5/6)ⁿ. The common guess that two dice give you a 2 in 6 chance of a six is wrong; it is 11 in 36, because the double six is counted only once.

The same formula answers "at least one 1", "at least one 20 on several d20s" and, with 1/2 in place of 1/6, "at least one head". For a d20, the chance of at least one natural 20 in n rolls is 1 − (19/20)ⁿ: 9.75% with two rolls, 22.6% with five, 40.1% with ten.

The d20 with advantage and disadvantage

A single d20 hits a target number k or higher with probability (21 − k)/20. With advantage you roll twice and keep the higher result, so you fail only if both rolls fail: the chance of success is 1 − (1 − p)². With disadvantage you keep the lower result, so both rolls must succeed: p².

Advantage is worth the most in the middle. At a target of 11 it turns a coin flip into a 75% shot, the same as a +5 bonus would. At the ends it is worth much less: a natural 20 goes from 5% to 9.75%, which is like a +1. Disadvantage mirrors this exactly. The dice roller has advantage and disadvantage buttons that roll the pair and mark which result counts.

Coins: exactly k heads in n flips

With n flips there are 2ⁿ equally likely sequences. The number of them with exactly k heads is the binomial coefficient "n choose k", so the probability is C(n, k) ÷ 2ⁿ. The last column adds up every outcome with k heads or more, which is the question behind "best of three" and "at least half".

Two of these rows surprise people. Exactly five heads in ten flips, the "expected" result, happens only 24.6% of the time; most series are a little lopsided. And ten heads in ten flips is 1 in 1,024, rare for a single try but not for a classroom: if 30 people each flip ten coins, the chance that at least one of them gets all heads is 1 − (1023/1024)³⁰, about 2.9%. The coin flip keeps a count of heads and tails across a session so you can compare a real series with the table.

Streaks: how long a run is normal

A run of five heads in five named flips is 1 in 32. But a series of flips contains many places where a run can start, and that changes everything. The table gives the chance that a series of n flips contains, somewhere inside it, a run of at least r identical results (heads or tails). It is computed by stepping through the flips and tracking the length of the current run.

In 100 flips a run of seven or more of the same face is more likely than not. The same logic applies to dice: in 100 rolls of a d6, the chance that some number repeats three times in a row is high, and a casino or a game app is not cheating when it happens.

Other dice and other questions

• Any single die: each face is 1 in s for an s-sided die, so 1 in 4 on a d4, 1 in 8 on a d8, 1 in 12 on a d12, 1 in 100 on percentile dice.
• Two dice, specific sum or higher: read the "or higher" column above. For dice with other sides, count combinations the same way: a 2d10 total of 11 is the most likely at 10 in 100.
• Rolling the same number on two dice of different sizes: a d6 and a d8 match with probability 6/48 = 12.5%, because only the six shared faces can match.
• Expected value: the average of one d6 is 3.5, of 2d6 is 7, of 3d6 is 10.5, and of a d20 is 10.5. Multiply the number of dice by (sides + 1)/2.
• Random numbers without dice: the random number generator picks any range with or without repeats, which is the same as a die with that many faces.

Frequently asked questions

Written by Tobias Kurth, designer. Updated October 20, 2026.