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.

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## Author: Joseph Nebus

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

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And Him Upstairs says something different so I shall go and rethink.

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1 in 3 for rolling 1 die is right

There are 216 ordered triples with 3 dice

For rolling 3, first die 2 is 1/6, or 36/216

Second is (1/6)*(5/6), or 30/216

Third for 3 2s is 1/6 * 150/216, or 25/216

Add 5 ways to roll 3 of a kind without 2s (22111, etc)

96/216, or 4/9 beats 1/3, or 3/9

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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

twothree-of-a-kinds.LikeLike

You’re a small bit off, but the spirit of your calculations is right.

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