Dice probability: the math behind every roll

An ordinary die is the simplest randomness machine there is: six faces, each equally likely. As soon as you roll two or more dice, patterns emerge that not everyone expects, and they make the difference in games like Yahtzee and craps.

One die: the basic probability

A fair die has six faces, and each face has exactly the same chance. So the probability of any specific number is:

• Probability per face: 1/6 ≈ 16.67%, for each of the six faces.
• Expected value (average): 3.5. That's 1, 2, 3, 4, 5, and 6 added together, divided by six.

Of course you'll never roll a 3.5, because there's no face with 3.5 pips. But roll often enough and your average creeps closer and closer to 3.5. That's what "expected value" means: the average you expect in the long run. Try it yourself with the online dice roller and keep track of your rolls for a while.

Probability with 2 dice: the sum of the pips

As soon as you roll two dice at once, you usually look at the sum of the pips. There are 6 × 6 = 36 possible outcomes (the first die can land six ways, and for each of those the second die can land six ways again). The sum runs from 2 through 12, but those sums are far from equally likely.

The reason: there is only one way to roll a 2 (1-1), but six different ways to make a 7. The more ways, the higher the probability. Here is the complete distribution:

The numbers form a neat pyramid: low at the edges, high in the middle. The sum 7 is the most likely, with 6 of the 36 ways, or 16.67%. The expected sum of two dice is accordingly exactly 7, which makes sense, since it's twice the 3.5 of a single die.

The probability of doubles

A double (two matching faces) can happen six ways: 1-1, 2-2, 3-3, 4-4, 5-5, and 6-6. That's 6 favorable outcomes out of 36 in total:

• Probability of any double: 6/36 = 1/6 ≈ 16.67%. Roughly one in six rolls with two dice is a double.
• Probability of a specific double (double six, for example): 1/36 ≈ 2.78%. There is only one way to roll exactly 6-6.

Is it a coincidence that the probability of any double (1/6) equals the probability of one particular face on a single die? Not quite: once the first die has landed, the second die has to "match", and that chance is exactly 1/6.

Snake eyes: the odds of rolling 1-1

Two ones is the double everyone knows by name. A pair of single pips looks like a pair of eyes staring back at you, and that's where snake eyes comes from. The math is the same as for any other specific double, because exactly one of the 36 combinations is 1-1:

• Probability of snake eyes: 1/36 ≈ 2.78%, about once in every 36 rolls of two dice.
• Odds against: 35 to 1. For every one way to roll it there are 35 ways not to.
• Over a run of rolls: 1 − (35/36)ⁿ. It takes 25 throws before you're more likely than not to have seen at least one, and even after 50 throws there's a one-in-four chance you still haven't.

Snake eyes is also the lowest sum two dice can make, which is why games give it a special role. In craps, a 2 on the come-out roll is one of the three "craps" numbers that lose a Pass Line bet on the spot. What a table pays for a bet on the 2 itself is a house decision printed on the felt, so read the layout. In backgammon it's the opposite: 1-1 is a windfall, because doubles let you move four times.

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The odds of rolling a 6 with more dice

Say you roll several dice and want to see at least one six. Many people figure: 16.67% per die, so 33% with two dice, 50% with three, and so on. That's wrong. You can't add probabilities together like that. Otherwise six dice would get you to 100%, and you can easily roll six dice without a single six.

The correct approach flips it around: first calculate the probability that you roll no six at all, then subtract that from 1. The probability of no six with one die is 5/6. With k dice, each landing independently of the others, you multiply that probability by itself k times. The formula becomes:

Probability of at least one six = 1 − (5/6)^k

The probability does rise, but ever more slowly: each extra die adds less than the one before. With six dice you reach 66.51%, well below the 100% you'd get by adding up 16.67% six times. You can check it yourself: set the online dice roller to six dice and count how often at least one six shows up.

Three dice: the 3d6 distribution

Add a third die and the pyramid turns into a bell. There are now 6 × 6 × 6 = 216 outcomes, the sum runs from 3 to 18, and there is no single peak any more: 10 and 11 tie for most likely, with 27 ways each. Between them they account for a quarter of every roll.

The expected sum is 10.5, three times the 3.5 of a single die. What changes is the shape. The four middle sums, 9 through 12, already cover about 48% of every roll, while the two extremes (3 and 18) together turn up just once in 108 rolls. Each die you add flattens the ends and fattens the middle further. With enough dice the sums form a bell curve.

Three six-sided dice is the oldest spread in role-playing games, written 3d6 in the notation the hobby uses. If that syntax is new to you, our page on dice notation explains it. The best-known variation shifts the curve on purpose: roll four dice, drop the lowest, and the average climbs from 10.5 to about 12.24, with the low sums almost disappearing. The full distribution is worked through on the 4d6 drop lowest page.

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Polyhedral odds: the d20 and the percentile pair

None of this is special to six faces. On any fair die with n faces, every face has probability 1/n and the average roll is (n + 1) ÷ 2. Those two formulas cover the whole polyhedral set:

Rolling "X or higher" on a d20

Role-playing games rarely ask for one exact number. They ask you to hit a target. On a d20 the faces X through 20 are the ones that succeed, so the probability is (21 − X) ÷ 20. Needing a 15 or higher gives you six winning faces out of twenty, or 30%. Needing an 11 or higher is exactly half. And a natural 20 is one face in twenty, or 5%.

Advantage: two d20, keep the higher

Several systems let you roll two d20 and take the better result. You can price that with the same trick as "at least one six": work out the chance that both dice miss, then subtract from 1.

• At least one natural 20: 1 − (19/20)² = 9.75%, nearly but not quite double the single-die 5%.
• Hitting a 15 or higher: 1 − (14/20)² = 51%, up from 30%.
• Hitting an 11 or higher: 1 − (10/20)² = 75%, up from 50%.

The gain is biggest in the middle. A second die adds the most when the first one was a coin flip, and almost nothing when the roll was nearly certain or nearly hopeless, which is why advantage helps most on a check that is hard but not impossible.

The percentile pair

Two ten-sided dice (a tens die reading 00 to 90, plus an ordinary d10 for the units) give a flat 1 in 100 for every reading, and the pair 00 and 0 is read as 100 rather than 0. So a "35% chance" in a rulebook means 35 of the 100 equally likely readings. How the pair is read, and the systems that treat it as a 0-to-99 range instead, is covered on types of dice.

Roll a d20 online →

The odds of rolling a Yahtzee

Five of a kind on the first throw is the easiest case to work out. Five dice land in 6⁵ = 7,776 different ways, and exactly six of those are five of a kind: all ones, all twos, and so on up to all sixes. That makes it 6/7,776 = 1 in 1,296, or about 0.077%.

A turn in Yahtzee gives you three throws, though, and you choose what to keep in between. Play it the obvious way (hold your biggest matching set, reroll the rest) and the chance of ending the turn with five of a kind is about 4.6%, or roughly one turn in 22.

You'll see that written both as "1 in 21" and as "1 in 22", which isn't a contradiction: the exact value is 1 in 21.7, and writers round it differently. Lower numbers circulate too (some published tables land nearer 1%) because they model a different rerolling strategy. The two figures to remember are 1 in 1,296 for a single throw, and a few percent for a full turn.

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Key concepts: independence & expected value

Three concepts explain almost all dice probabilities. Understand these and you can work everything out yourself from now on.

Independent events

Every roll is completely independent of the last. A die has no memory: whether you just rolled three sixes in a row or have never seen a six at all, the probability of the next six stays 1/6. The idea that an outcome is "due", or that a long streak of sixes must be "compensated" by a low roll, is called the gambler's fallacy.

Expected value

The expected value is the average you expect in the long run: 3.5 per die, 7 for two dice, 10.5 for three. You almost never roll it exactly, but over thousands of rolls the average creeps closer and closer to it. This is the law of large numbers at work.

This is easier to show than to explain. Set the roller going, keep a running tally, and watch the average settle down by itself. The classroom dice roller is set up for that kind of lesson. With the free Android app you can throw the dice onto a TV, so a whole table or classroom reads the same roll at the same moment instead of crowding around one laptop.

Combined probabilities through multiplication

Want the probability that several independent things all happen? Then you multiply their probabilities. The probability of two sixes in a row, for example, is 1/6 × 1/6 = 1/36 ≈ 2.78%. That's why the "at least one six" trick above worked: (5/6) × (5/6) × … for each die gives the probability of no six.

What it means for dice games

These probabilities drive the strategy in every dice game.

• The magic 7 in craps. In craps, everything revolves around the 7. It comes up most often (16.67%), which makes it your friend on the come-out roll but your biggest enemy later in the round. The entire tension of the game flows from that pyramid shape.
• The bonus hunt in Yahtzee. In Yahtzee you roll five dice per turn, so the odds of at least one of a given number are high. Understand the "at least one" formula and you'll know better when to chase the upper-section bonus and when to sacrifice a box instead.
• Aiming for the middle. In almost every game with two dice, most of the probability sits around 6, 7, and 8. Betting on the extremes (2 or 12) is exciting, but rare. That's why the middle numbers are the easiest to cross off in Qwixx.

For the rules themselves, see the complete overview of dice games. For each game you'll see how the odds shape the tactics.