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.
nebusresearch 
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
LikeLike
And Him Upstairs says something different so I shall go and rethink.
LikeLike
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
LikeLike
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.
LikeLike
You’re a small bit off, but the spirit of your calculations is right.
LikeLike