Three doubles in a row

Your opponent has just rolled three doubles in a row, turning around a game you thought was going well.

You might do a quick calculation: one in six, multiplied by one in six, multiplied by one in six. That is one in 216. It sounds like something you should hardly ever see.

The calculation is right for three rolls chosen before the first one is thrown. During a game, the question often comes after the first double: will the next two rolls also be doubles?

The first double has already happened

If your opponent has already rolled a double, you are now waiting to see whether they will roll another.

The chance of a double on their next roll is still one in six. The previous double makes another neither more nor less likely.

And the chance that their next two rolls are both doubles is one in 36. If that happens, you’ll have watched three doubles in a row.

Once the first double has happened, you only need to predict the next two rolls, which is why the chance is one in 36 rather than one in 216.

All three statements are true because they begin at different moments.

A session gives the streak somewhere to start

There is another reason these runs show up more often than one in 216 might suggest. You don’t normally sit down, watch exactly three rolls, and stop.

A run can begin on the first roll, the second or the twenty-seventh, so a longer session gives it more opportunities to appear.

Across 100 independent rolls, the chance of seeing at least one run of three or more doubles is about 31.9%. Roughly one in three sets of 100 rolls will contain one.

That doesn’t mean the next three rolls have become any more likely to be doubles. It means we have given the event many more opportunities to happen.

Even two doubles together are easy to underestimate. In 100 rolls, the chance of at least one back-to-back pair is about 91.3%.

Here, count the rolls in the sequence you mean. If the story is “my opponent rolled three doubles in a row”, follow that player’s own turns. Your rolls in between don’t interrupt their sequence. And 100 rolls means 100 rolls, not 100 games.

Pick the rolls first, then look at the whole set

The example below lets you try both questions on the same dice. The first three rolls are marked before anything happens. Their chance of all being doubles is one in 216.

Then look through the whole set. A streak anywhere counts for the second question. You can roll one at a time or fill the rest at once.

A set with no streak is ordinary: if 31.9% of 100-roll sets contain a run of three or more doubles, most do not.

What the streak tells you

To judge how unusual three doubles were, even if they cost you a race, specify where the calculation starts: before three chosen rolls, after the first double, or across the whole session.

This explains why a streak can occur with fair dice. It doesn’t establish that any particular set of online dice is fair; checking that takes evidence about how those dice are generated.

For another surprising effect, two equally likely patterns can have very different chances of appearing first. We also explain how CUBE Backgammon generates its dice and lets you replay recorded rolls.

Where the numbers come from

We assume fair, independent rolls. A double has probability 6/36 = 1/6. For three rolls picked in advance, multiply 1/6 three times. The 100-roll figures are calculated by keeping track of how many doubles are currently at the end of each possible sequence, excluding sequences that have already reached the target streak. Some groups overlap: rolls 1–3 and rolls 2–4 have two rolls in common. Treating those groups as independent would give the wrong answer.

The interactive example uses the browser’s cryptographic random-number generator to simulate fresh rolls. The percentage beside it is calculated directly, not estimated from that one sample. For the broader mathematics, see Runs in coin tossing: a general approach for deriving distributions for functionals, by Lars Holst and Takis Konstantopoulos.