05 October 2009

Farkle Probabilities

I've been playing Farkle on Facebook, so I started getting curious about the probabilities involved with the game. I'm not the only one doing this, as I've found several others doing much the same (1,2,3). You can also play Farkle at TADMAS. It took me a quite a bit of time tinkering with a spreadsheet until I understood how to think about this. I'm already up past my bedtime tonight, so I'm going to spare you the gory details and get right to the results. This post will focus on the number of dice you roll and the number of times you roll them. I haven't completely figured out the scoring distribution yet, so that may be a follow-up post.

The following table gives the probabilities for how many dice will remain after you remove all the dice which score points (but you might choose not to remove all). For instance, when you roll 6 dice, there is a 2.31% chance of not scoring at all, or a "Farkle" (marked in red), and 15.43% chance that you will be able to score exactly one die (a 1 or a 5). The percentages (marked in green) are the probabilities that all remaining dice can be scored, thus gaining a "new roll".

If you score one die and re-roll the remaining 5 dice, and there is (for example) a 30.86% chance that there will be 2 (three Sir) 3 dice left (and therefore the other two are 1's or 5's).
As you play, there is a choice to not score all of the dice. You might do this in hopes of getting a better throw on the next try, and you might use this table to consider the risk of choosing to re-roll 1's and 5's, instead of scoring them immediately.

That first table looked the the game from the "one roll at a time" perspective. The next table is a little more complicated because it "looks ahead" at what will happen if you score all the dice you can and keep rolling until you Farkle or get a new roll. Win/lose percentages marked in red and green are as before. Numbers in shaded gray are intermediate probabilities used in my calculations and have no simple interpretation.

These (red and green percentages) are conditional probabilities for what might happen. If you roll 6 dice, AND score one of those, AND roll the 5 remaining dice, there is a 1.19% chance of a Farkle.

My final table presents some some further conditional probability calculations, and gives the overall probabilities of either Farkle-ing or getting a new roll, if you score all possible dice and re-roll all that remain.

Starting with 6 dice, there is a 68.63% of Farkling before getting a new roll, assuming you choose not to "cash-in" your points first. If you want to think about the possibility of getting several re-rolls (thus scoring a large number of points), you can look at each group of 6 dice as a geometric series - a probability distribution which I have mentioned a few times before.

You can use these tables to inform yourself about the risk of Farkling as you play the game. This might help you understand the game, but by itself it probably won't help you achieve a high-score to beat all your friends. To do that will require an understanding of the relationship between the risk of losing your points versus the probability of achieving your target high-score. When I figure that out, I'll let you know.
[some revisions, 3/26/2011]
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