Roll 2 dice: an online two dice roller

How to use the two dice on this page

The tray above holds two standard six-sided dice, written 2d6 in role-playing games. You can throw them in two ways:

• The Roll button: both dice are launched with a random push, tumble across the tray and come to rest on their own.
• Drag and drop: pick the dice up with the mouse or your finger, move them and let go. The speed of your release becomes the force of the throw, much like rolling real dice by hand.

Once both dice stop, the page reads the top face of each, shows the two numbers and adds them up. If both dice show the same number, a doubles alert appears, which is what matters in games such as backgammon and Monopoly.

Underneath, a tally builds up over the session: how many times each total from 2 to 12 has come up, one bar per total, with a marker showing what probability would lead you to expect. You can watch the shape of your results drift towards the theoretical one roll by roll.

The 36 combinations of two dice and the table of totals

Each die has 6 faces, so two dice give 6 × 6 = 36 possible outcomes. To count them properly you have to tell the dice apart, as if one were red and the other blue: a 1 on the red die with a 2 on the blue is a different outcome from a 2 on the red with a 1 on the blue, even though both total 3. With fair dice, each of those 36 outcomes is equally likely, with a probability of 1/36.

The totals are not equally likely, because each one can be made in a different number of ways. Here is the full table:

The last column tells you how often each total turns up in the long run: a 7 once every 6 rolls, a 2 or a 12 once every 36. There are 11 possible totals, from 2 to 12.

If you ignore the order, there are 21 distinct pairs: 6 doubles and 15 pairs of different numbers. That count tells you which pairs exist, but it is no use for working out probabilities, because the pairs are not equally likely. Each double can only happen one way out of 36, while each pair of different numbers can happen two ways. That is why a 1 and a 2 (in either order) is twice as likely as double one.

Why 7 is the most likely total

Seven is the only total you can still make whatever the first die shows. If the first die lands on 1, the second needs a 6; on 2, a 5; on 3, a 4; and so on up to 6, which needs a 1. There is always exactly one face on the second die that gives 7, so the probability is 6/36, or 1/6 (16.67%).

Every other total rules out some values of the first die. To make 8, a 1 on the first die is no good, because the second would have to show 7. For 12, only a 6 on each die will do. The further a total is from 7, the more first-die values are ruled out, which is why the table drops by one step at a time towards each end and forms a symmetrical triangle: 6 and 8 are equally likely, and so are 5 and 9, 4 and 10, 3 and 11, and 2 and 12.

Seven is also the average total. One die averages 3.5, so two dice average 7. The standard deviation is about 2.42, and two rolls in three land between 5 and 9 (24 of the 36 combinations, 66.67%).

Being the most common total does not make 7 a near certainty: five rolls in six give something else. It simply comes up more often than any other single total.

Probability of doubles, of rolling a six and other outcomes

Any two dice calculation comes down to counting how many of the 36 outcomes match what you are after. The questions people ask most often:

Over several rolls, the chance of seeing something at least once rises more slowly than most people expect. It is 1 minus the chance of never seeing it: for doubles, 1 − (5/6) to the power of the number of rolls; for a 12, 1 − (35/36) to the power of the number of rolls.

Even 36 rolls do not guarantee a 12: there is still a 36% chance of not seeing one. And to see every total from 2 to 12 at least once takes about 61 rolls on average, mostly because of the 2 and the 12.

Games played with two dice

Plenty of classic board games use a pair of six-sided dice. These are the dice rules of the best known ones, which is all you need to use this page in place of physical dice.

Backgammon

Each turn you roll two dice and move your checkers by the two numbers, either with one checker or with two. A double is played four times instead of twice: 5-5 gives four moves of five points each. If only one of the two numbers can be played, you must play the higher one. To start the game, each player rolls a single die; the higher number goes first and moves using both numbers just rolled. On a tie, both roll again. The doubling cube that comes with the board is not a die: it is marked 2, 4, 8, 16, 32 and 64, is never rolled and only keeps track of the stakes.

Monopoly

You move as many squares as the two dice add up to. Roll a double and you move, then roll again. Roll three doubles in a row in the same turn and you go straight to Jail without making the third move. The chance of three doubles in a row is 1 in 216, or 0.46%.

Catan

At the start of each turn you roll two dice and add them up. Every hex marked with that number produces resources for the settlements and cities next to it (one card per settlement, two per city). On a 7 nothing produces: anyone holding more than 7 cards discards half, rounding down, and the player who rolled moves the robber. That is why hexes numbered 6 and 8 are the most productive: between them they come up 10 times in 36, almost 28%, while 2 and 12 together come up only 2 times in 36.

Craps

The classic casino dice game. On the first roll of a round, the come-out roll, a 7 or an 11 wins for anyone on the pass line, with a probability of 8/36 (22.22%). A 2, 3 or 12, known as craps, loses (4/36, 11.11%). Any other total, 4, 5, 6, 8, 9 or 10, becomes the point, and the shooter has to roll it again before a 7 to win. Two come-out rolls in three set a point. Players have their own names for some rolls: snake eyes for double one, boxcars for double six, and yo for 11, so it is not misheard as seven.

The game of the goose

In Italy, the gioco dell'oca is played with two dice, and many versions elsewhere use two as well, although some are played with one. Where two dice are used, a well known opening rule sends a first roll of 6 and 3 straight to square 26 and a first roll of 5 and 4 to square 53. The rules vary a great deal between countries and editions, so check the ones printed with your board.

Ludo and other race games

Ludo is normally played with a single die, as are Snakes and Ladders and most children's race games, so for those a one-die roller is the better fit. Its Spanish cousin, parchís, also uses one die in its classic form, though a two-dice version has become popular for online play.

Galileo, the Chevalier de Méré and the birth of probability

Two of the best known problems in the history of probability came out of dice games, and both are solved with the same sort of counting as the table above.

Galileo and the three dice

Early in the 17th century, Galileo Galilei wrote a short piece, known as Sopra le scoperte dei dadi, to settle a question from gamblers. With three dice, 9 and 10 can each be made from six combinations (for 9: 1-2-6, 1-3-5, 1-4-4, 2-2-5, 2-3-4 and 3-3-3), yet experience suggested that 10 came up slightly more often. Galileo solved it by counting ordered throws: of the 216 possible outcomes with three dice, 27 total 10 and only 25 total 9. The unordered combinations do not carry equal weight: 3-3-3 can only happen one way, whereas 1-2-6 can happen six ways. The question is traditionally said to have come from the court of the Grand Duke of Tuscany, Galileo's employer, though the text itself does not name whoever asked.

It is exactly why, with two dice, a 1 and a 2 is twice as likely as double one.

The Chevalier de Méré, Pascal and Fermat (1654)

Antoine Gombaud, known as the Chevalier de Méré, compared two bets. The first, getting at least one six in four rolls of a single die, is favourable: 1 − (5/6)⁴ ≈ 51.77%. By the rule of proportion gamblers used at the time, the second should have been favourable too: double six is six times less likely than a six, so six times as many rolls, 24, ought to be enough. The correct calculation says otherwise: the chance of at least one double six in 24 rolls of two dice is 1 − (35/36)²⁴ ≈ 49.14%, just under a half. It takes 25 rolls to get above 50% (50.55%).

Blaise Pascal reported de Méré's complaint in a letter to Pierre de Fermat dated 29 July 1654. Their exchange of letters that year, which also dealt with the problem of points (how to split the stakes when a game is interrupted), is generally seen as the starting point of modern probability theory.

The law of large numbers: why your chart does not match the theory

The probabilities in the table describe what happens in the long run, not in a short session. Over 36 rolls you would expect six 7s, one 2 and one 12, but it is perfectly normal to get three 7s, or ten, and no 12 at all. The law of large numbers says that the share of rolls landing on each total gets closer to its probability as the number of rolls grows. It says nothing about the next roll.

This table shows, for the total of 7, the range the count falls in during 95 out of 100 sessions (worked out with the binomial distribution):

After 36 rolls, anything from 2 to 11 sevens is normal. After 1,000, the percentage sits between 14.4% and 19%, close to the theoretical 16.67%. Two points help when reading the chart:

• The range narrows slowly. To halve it you need four times as many rolls. That is why a few dozen rolls look almost random, a couple of hundred start to show the triangle, and several thousand put the bars very close to the markers.
• Dice have no memory. If 7 has not come up for a while, its chance on the next roll is the same as always, 1 in 6. The chart drifts towards the theory because early streaks carry less and less weight among many rolls. The dice do not make up for past results, and believing they do is known as the gambler's fallacy.

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