Yahtzee Probability Calculator

How it works

Exact dynamic programme over the 252 distinct rolls of five dice: chance(dice, r rolls left) = max over which dice to keep of the average chance over every outcome of the re-rolled dice with r − 1 rolls left; with no rolls left the chance is 1 if the dice score the goal and 0 if not. For a Yahtzee this equals the Markov chain over the number of matching dice, P(Yahtzee) = [T^rolls]₍start,5₎, which gives 2,783,176 ÷ 60,466,176 = 4.60% for a full turn.

Each roll of a Yahtzee turn leaves you with a set of dice, and because the dice are fair, the chance of reaching any goal can be counted exactly. The calculator lists every outcome of the dice you re-roll and assumes that after each roll you keep the dice that give the best chance of the goal. For a Yahtzee that means keeping the largest matching group, which gives the famous 4.60% for a full three-roll turn, or about 1 in 22. The same method works for the other boxes: a full turn aimed at a large straight succeeds 26.1% of the time, a small straight 61.5%, a full house 36.3%, four of a kind 29.1% and three of a kind 74.3%. Type in the dice you just rolled to see which ones to keep and how much each choice is worth, or set how many matching dice you hold and how many rolls remain. The upper-section goal shows the chance of rolling at least, say, three sixes, which is par for the 63-point bonus. The tables show every goal with 1, 2 or 3 rolls and the classic Yahtzee odds by position. These are single-goal odds; the best play in a real game also weighs the score and which boxes are still open.

Worked example

Holding three of a kind with two rolls left, the chance of finishing a Yahtzee is 9.34%: 1/36 on the next roll straight away, plus 10/36 × 1/6 via four of a kind, plus 25/36 × 1/36 by staying on three and hitting both on the last roll. From nothing, a full three-roll turn succeeds 4.60% of the time, 1 in 21.7, so a player who chases it every turn expects 0.60 Yahtzees per game and gets at least one in 46% of games. If your first roll is 2 3 4 6 6 and you want a large straight, keep 2 3 4 6 and re-roll one die: only a 5 completes it, so the chance over two rolls is 1 − (5/6)² = 30.56%, better than keeping just 2 3 4 (27.78%). Starting from 2 3 4 5, a 1 or a 6 will do, so the chance is 1 − (4/6)² = 55.56%.

Frequently asked questions