Five dice odds made simple: your real chances with three rolls

What are the odds on a single roll?

Five ordinary dice can land in 6 x 6 x 6 x 6 x 6 = 7,776 equally likely ways. Counting how many of those make each pattern gives the chance of rolling it straight away. We counted these ourselves; the pattern counts (five of a kind, four of a kind, full house, all different) match DataGenetics.

So a first roll shows at least a pair about nine times in ten, and a ready-made full house or large straight only about once in every 25 to 33 rolls.

Five dice odds over three rolls

Holding dice and rerolling the rest is where the chances open up. Keep your most common number each time and reroll everything else, and five of a kind in one turn comes out at about 4.6%. Cornell, DataGenetics and the Wizard of Odds all reach that figure using a Markov chain, a way of tracking how likely you are to move from “I have a pair” to “I have three” with each roll.

The same Cornell model shows two other useful things when you are chasing one number:

• You finish the turn with at least three of a kind just under 75% of the time.
• You finish with four of a kind or better about 29% of the time.

That is why chasing a high number such as sixes is usually a sound plan: even when it misses, three or four of them is the likely result.

One caution: real players often switch to a straight or take an early full house, so in real games five of a kind comes up a little less often than 4.6%. The 4.6% assumes you never change your mind.

Should you keep an open or an inside straight?

Say you hold four dice that are one short of a straight and you have two rolls left with one die.

• Open ended (for example 2, 3, 4, 5, where a 1 or a 6 finishes it): two numbers work, so your chance over two rolls is 5/9, about 56%.
• Inside (for example 1, 2, 4, 5, where only a 3 works): one number works, so your chance falls to 11/36, about 31%.

Cornell works through both cases. The lesson: when you can choose, keep the run with two ways to finish.

Quick tips that follow from the numbers

1. A pair is normal, so treat it as a starting point. Three of a kind on the first roll (about one in five) is the real head start.
2. Chasing one number is forgiving. Even when five of a kind misses, you usually land three or four of them.
3. Prefer open ended straights. Two ways to finish beats one by a wide margin.
4. Plan around likely results. One turn in about 22 is lovely when it happens, but it is the exception.
5. Think about what is still open. A roll is worth less once the boxes it suits are already used.

Questions people ask

Where Dice Charms fits, and where it does not

Dice Charms is our dice roguelite for Android. It starts from five dice, three rolls and a scorecard of poker style hands. Each round has a target to beat, and between rounds a shop sells charms, special dice with new faces and stamps that change how you score. Every card ends with a guest who brings a house rule.

The single-roll odds above still apply to plain dice in Dice Charms, and give a useful instinct for which dice to hold. Once you add charms, special dice and guest rules, though, the maths shifts, and that is the point of the game: you are bending the rules on purpose. For exact probabilities under a changed rule, a worked source like Cornell’s is the better tool.

Dice Charms is coming soon to Google Play. Grade 1 is free to play; Grades 2 and 3 are a one-time unlock. There are no ads, no subscriptions and no timers, and it plays fully offline. Questions go to Email us.

Sources

1. Probability Games lessonsCornell University Department of Mathematics · pi.math.cornell.edu
2. Five of a kind probabilityDataGenetics · datagenetics.com
3. Ask the Wizard #335Wizard of Odds · wizardofodds.com