A Quick Yahtzee Puzzle

My friend ChefMongoose pointed out this probability question. As with many probability questions, it comes from a dice game. Here, Yahtzee, based on rolling five dice to make combinations. I’m not sure whether my Twitter problems will get in the way of this embedding working; we’ll see.

The question:

Probability help please! You are playing Yahtzee against your insanely competitive spouse. You have two rolls left. You’re trying to get three of a kind. Is it better to commit and roll three dice here? Or split it and roll one die? — Christopher Yost.

Of the five dice, two are showing 1’s; two are showing 2’s; and there’s one last die that’s a 3.

As with many dice questions you can in principle work this out by listing all the possible combinations of every possible outcome. A bit of reasoning takes much less work, but you have to think through the reasons.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

6 thoughts on “A Quick Yahtzee Puzzle”

  1. If you hold onto the two pairs and roll one die, you need either a 1 or a 2 – so there’s a 2 in 6 (or 1 in 3) chance of making it.

    If you roll 3 dice to match the remaining pair, you have a 1/6 chance of rolling that number on one die, times 3 for the 3 dice, giving you 3/6 or 1/2 chance.

    Which means my strategy changes right now


        1. And yeah, that’s how to finish the calculation off. Rolling three of a kind on the three rerolled dice is the catch that I had overlooked in doing the calculation.

          And it really shakes my intuition that rolling three dice is so more likely to give three-of-a-kind than just rolling one die when you’re already two-thirds of the way to two three-of-a-kinds.


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