# What is the probability of getting a Yahtzee within $n$ rolls?

Say there is the standard situation in Yahtzee with $$5$$ $$6$$-sided dice. The player tries from the outset to get a Yahtzee. The player is given the conventional liberty of choosing which dice to re-roll in the next roll, as well as the new liberty of having as many rolls as necessary to achieve a Yahtzee, instead of the normal $$3$$ rolls. Obviously the probability of getting a Yahtzee within $$1$$ roll is $$\frac{1}{1296}$$, and the probability of getting a Yahtzee within $$n$$ rolls approaches $$1$$ as $$n\to \infty$$. But what happens in between, as $$n$$ goes from $$2$$ to $$\infty$$?

First we have to describe the optimal strategy for getting a Yahtzee. It's not hard to see that it's given by:

• If none of the current dice match, reroll all $$5$$ dice
• If $$2$$ match, reroll the other $$3$$. (If there are two twos-of-a-kind, choose one arbitrarily -- say, the higher one -- to keep.)
• If $$3$$ match, reroll the other $$2$$.
• If $$4$$ match, reroll the other $$1$$.
• If $$5$$ match, you're done.

So it makes sense to describe the game as being in one of the above $$5$$ states. You'll need to compute the probabilities of moving from any state to any other state. Let $$P$$ be the matrix where $$P_{ij}$$ ($$i$$th row, $$j$$th column) is the probability of moving from state $$j$$ to state $$i$$. (Note that the matrix must be lower triangular -- can you see why?) The game starts in state $$1$$ (no matches if you haven't rolled any dice yet), so let $$v$$ be the column vector with a $$1$$ in the first entry and the others $$0$$. Then $$P^nv$$ gives the probabilities of the game being in each state after $$n$$ rolls. In particular, you're interested in the fifth entry of $$P^nv$$.

• Alternatively, for the first one, you could "choose one arbitrarily -- say, the higher one -- to keep" and roll the other five. Commented Apr 13, 2020 at 0:55
• @RayButterworth Yup! I chose my way to make it consistent with the first roll being in state $1$ Commented Apr 13, 2020 at 1:28
• I like that reasoning. You'd make a good computer programmer. Commented Apr 13, 2020 at 13:26