2 Dice Roller

What this two dice roller does

This page rolls two six-sided dice, nothing else. Each click throws both dice and shows the two faces, the sum, and whether the pair is doubles. A modifier field adds a flat number to the total, which is useful for games that roll 2d6 plus a bonus, and the rolls per click field throws a whole batch so you can build a sample fast.

Under the dice there are two tables. The first repeats the theory: 36 pairs, 11 possible sums, 7 the most common. The second is your own session, with a count for every sum, the share you saw, and the share the math predicts. Nothing is rounded to make your rolls look tidy, so a short session will wander away from the expected column and that is the point.

The log keeps the last dozen throws in order, newest first, so you can replay a hand or check a disputed roll. Both dice are fair, each face has a 1 in 6 chance, and the two throws are independent. Everything runs in your browser, so nothing is uploaded or stored.

The 36 outcomes behind the 2 to 12 spread

Two dice do not produce 11 equally likely sums. They produce 36 equally likely pairs, and the sums pile up on top of each other in the middle. Die one shows 1 and die two shows 1 in exactly one way, so a sum of 2 has a single route. A sum of 7 has six: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. Order matters here, which is why 3+4 and 4+3 count twice.

Read the table as a recipe. The ways column adds to 36, the probabilities add to 100 percent, and the cumulative column tells you how often a sum or lower comes up. A 7 or lower happens 21 times in 36, or 58.3 percent, which is a little better than a coin flip.

The symmetry is not a coincidence. Whatever die one shows, die two is just as likely to be low or high, so the shape mirrors around 7. Every sum below 7 has a twin above it with the same count: 6 and 8 both have five ways, 5 and 9 both have four.

Why 7 wins the race, and by how much

Seven is the most likely sum because it sits in the middle of the range, where the most pairs overlap. Six of the 36 pairs make it, so the probability is 6 divided by 36, which is 1 in 6, or 16.7 percent. The extremes lose badly: 2 and 12 have one pair each, 2.8 percent, so a 7 is six times as likely as snake eyes.

The average sum is 7. Each die averages 3.5, and adding two averages gives 7, which matches the peak. The spread is measured by a standard deviation of about 2.415 for the sum of two independent fair dice, so most rolls land within a couple of points of 7. Sums from 5 through 9 hold 24 of the 36 pairs, 66.7 percent, which is the band you should expect to see most often at a table.

None of this makes a 7 likely on the next roll. Each throw is independent, the dice have no memory, and the 16.7 percent applies to every throw separately. A table that has not seen a 7 in eight rolls is not due for one.

Doubles, snake eyes and at least one six

Doubles are the six pairs where both dice match: 1+1, 2+2, 3+3, 4+4, 5+5 and 6+6. That is 6 of 36, or 16.7 percent, the same share as a 7, and it is why game rules that punish doubles feel like they fire often. Snake eyes and boxcars are each one pair out of 36, 2.8 percent, and the pair of them together is 5.6 percent.

At least one six is a different question, and the easy way in is the opposite. The chance that a single die misses the six is 5 in 6, so both dice miss it with probability 25 in 36. Subtract from 1 and 11 of the 36 pairs contain a six, which is 30.6 percent. That covers 6+1 through 6+6 and 1+6 through 5+6, with 6+6 counted once.

Two related numbers worth keeping: exactly one six is 10 of 36, 27.8 percent, and the chance of no doubles at all is 30 of 36, 83.3 percent. Games that reroll on doubles, or that treat doubles as a bonus, are working with a one in six event, not a rare one.

The craps come-out roll, counted

Craps is the game most people meet two dice through, and the first throw of a round, the come-out roll, has three possible outcomes. A 7 or 11 wins the pass line bet outright, a 2, 3 or 12 loses it, and any other total becomes the point that must be repeated before a 7 shows up. The Nevada Gaming Control Board publishes rules of play for approved games, and its Crapless Craps rules of play describe the standard game by listing what that variant changes: there, 7 is the only come-out winner and 2, 3, 11 and 12 all become points.

The counts fall straight out of the 36 pairs. Winning totals 7 and 11 have 6 and 2 pairs, so 8 of 36, 22.2 percent. Losing totals 2, 3 and 12 have 1, 2 and 1, so 4 of 36, 11.1 percent. The remaining 24 pairs, 66.7 percent, set a point and send the round into a second phase.

Those last three rows are conditional. Once a 6 is the point, only a 6 or a 7 matters, so the chance of making it is 5 out of 11.

100 rolls: what to expect and what counts as noise

Run 100 throws and the expected count for each sum is 100 times its probability: about 17 sevens, 14 sixes, 14 eights, and 3 twos. Few sessions land on those numbers exactly. The count of sevens in 100 rolls behaves like a binomial with 100 trials at 1 in 6, which gives a standard deviation near 3.7, so anything from roughly 9 to 24 sevens sits inside two standard deviations and needs no explanation.

That example session has 17 sevens and 14 sixes, close to plan, while 5 elevens and 9 tens drift a little. Sum the difference column and it lands near zero, which is what a fair pair of dice does over time. A table of 20 rolls is too small to read: the same dice can show three 12s in 20 throws and still be fair.

The widget uses the browser's built-in random function, which is fine for games and demonstrations but is not a cryptographic generator. Software that needs unpredictable numbers for security uses a different class of tool, described in NIST SP 800-90A Rev. 1.

Monopoly jails, classrooms and checking a loaded die

The best known doubles rule in board games is Monopoly. Hasbro's official Monopoly rules state that a player who throws doubles three times in succession moves straight to Jail, and the roll also decides how far you move, which is why doubles feel like a gift until the third one. Three doubles in a row is 1 in 6 cubed, 1 in 216, or 0.46 percent of turns.

In a classroom, two dice are the cheapest way to show a distribution that is not flat. Hand out the 36 pair table, have students tally 60 rolls, then compare the tally to the theory column. The middle sums win almost every time, and the rare 2 and 12 make the lesson stick better than a lecture.

Loaded dice are harder to catch than people think. Compare a few hundred rolls against the expected counts before calling a die crooked, and remember that MathWorld's dice entry derives the count of ways for any number of dice, so the target numbers are easy to check.