What counts as a Yahtzee
In the dice game Yahtzee, a player rolls five dice and may reroll any of them up to two more times in a turn, so there are at most three rolls. A Yahtzee is five dice showing the same number. This article works out how likely that is. The figures are illustrations of probability for a fair set of dice, not advice about how to play for any prize.
In one roll
Five dice give 6^5 = 7,776 equally likely outcomes, counting each die separately. Five of a kind can be made in 6 ways, one for each number (all ones, all twos, and so on). So
P(five of a kind in one roll) = 6 / 7,776 = 1 / 1,296, about 0.077 percent.
Another way to see it: the first die can show anything, and each of the other four must match it, which has chance (1/6)^4 = 1/1,296. The two methods agree.
Rerolling a few dice
The chance rises a lot when you may reroll. The odds depend on what you hold, so here are three cases, each with the number of rerolls stated.
Holding four of a kind, one reroll left. You roll one die and need it to match, so the chance is 1/6, about 16.7 percent.
Holding four of a kind, two rerolls left. You fail only if you miss twice, with chance (5/6)^2 = 25/36. So the chance of success is 1 - 25/36 = 11/36, about 30.6 percent.
Holding three of a kind, one reroll left. You roll two dice and both must match: (1/6)^2 = 1/36, about 2.8 percent.
Holding three of a kind, two rerolls left. Take the first reroll of two dice. There are three cases:
- Both match, chance 1/36. You are done.
- Exactly one matches, chance 2 x (1/6) x (5/6) = 10/36. You now hold four, with one reroll left and a 1/6 chance of finishing.
- Neither matches, chance 25/36. You hold three still, with one reroll left and a 1/36 chance of finishing.
The total is 1/36 + (10/36)(1/6) + (25/36)(1/36). In units of 1/1,296 these are 36 + 60 + 25 = 121, so the chance is 121/1,296, about 9.3 percent. That assumes you keep any newly matching dice, which is the natural thing to do.
| Position | Rerolls left | Chance of reaching five of a kind |
|---|---|---|
| Four matching | 1 | 1/6 (16.7%) |
| Four matching | 2 | 11/36 (30.6%) |
| Three matching | 1 | 1/36 (2.8%) |
| Three matching | 2 | 121/1,296 (9.3%) |
Starting from nothing
Now take a whole turn from the opening roll, using a simple approach: after each roll, keep the face that appears most often (the largest group), reroll the rest, and use all three rolls. Working through every case of the opening roll gives a total chance of about 4.6 percent, which is roughly 1 turn in 22. I obtained this figure by a systematic enumeration of the cases rather than a short formula, so treat it as a rounded value, and note that it applies to this simple approach. Real players also decide based on the scoring card, and other choices will give different numbers.
Even so, the figure shows that a Yahtzee in one turn is an uncommon event. Across many turns it still happens, and a game of 13 turns has plenty of chances, but it is far from guaranteed. A rough check: if each turn had a 4.6 percent chance, the chance of at least one in 13 turns would be 1 - (0.954)^13, about 45 percent. That calculation treats turns as independent and always aiming for a Yahtzee, which no real player does, so it is only an illustration of the complement rule and the effect of many tries.
What this tells you about dice
Two facts stand out. First, holding dice matters far more than rolling everything again: being at four of a kind makes success much likelier than starting with three. Second, the chance of reaching a rare outcome is built from many small paths, which is why careful counting beats intuition. The method used here, branching on what happens at each step, is the idea behind probability trees.
Remember also that dice are independent. Failing to get a Yahtzee in earlier turns does not make one more likely later, and a run of near-misses says nothing about the next roll, a form of the gambler’s fallacy. Expect the usual variation: a player can go many games without one, and another can get two in a game.
A note
Yahtzee is a game of chance and skill played for fun, and the numbers here are about understanding the dice. If you ever find that games involving money are causing you harm, a local support service can help.