I recently started playing the game of Farkle on Facebook. The basic decision is whether to "pick up your dice" and thereby end your turn, or throw again and continue your turn. If you use all six of your dice to score, you can then roll again (with all six dice) and continue that same turn. It is this decision which has aroused my curiosity; i.e., what are the odds of farkling when rolling all six dice?
First of all, we note there are 6 to the 6th power possible rolls of the 6 dice, or 46,656 possible rolls. We seek to determine how many of these rolls are farkles. The number of rolls in which neither a 1 nor a 5 appears is 4 to the 6th power, or 4,096. This already gets us to less than 10% farkles, but this is not the end of the inquiry, because some of those 4,096 will score in other ways, i.e., 3 of the same number, or 3 pairs.
We are only interested in ways of scoring which don't involve a 1 or a 5, since we have already ruled those rolls out. Let's take 6's, keeping in mind that at the end we will multiply by 4 to cover the other 3 numbers that aren't 1 or 5. There are 20 ways of rolling three 6's (combinations of 6 things taken 3 at a time). Those other 3 dice can be rolled 3 cubed ways, or 27, but then we subtract for the 3 times they will all be the same number, leaving us with 24. So the total rolls for three 6's is 20 x 24, or 480.
For four 6's we have 15 combinations, with the other two dice having 9 possibilites, for a total of 15 x 9, or 135. For five 6's, There are 6 x 3 = 18 ways. And for all six 6's, of course only one way.
Adding these up, we have 634 ways of throwing three or more 6's in a roll, and multiplying by four gives us 2,536. Subtracting this number from the 4096 yields 1,560, which is .033 of the total. Thus, one will farkle about one out of every thirty times rolling six dice.
The actual odds are even less, since I have not taken into account the possibility of rolling three pairs. (Note the straight has already been excluded, since it will have a 1 and a 5 in it.) However, the 1 in 30 should be solid enough to inform our rolling decision.
Say we have a straight, which scores 1,500 points. By rolling our expected loss would be 1/30 of that, as that is how often we will farkle and lose those points. This comes to an expected loss of 50 points. In my experience the average score on a turn is about 300, so the expected gain would be 300, making rolling an easy decision. However, the 300 average score is achieved through much more daring play than one would be willing to make when risking 1,500 points, so I am going to say the expected gain is more like 150-200 instead of 300 in this situation. (Or, to put it another way, you will likely only roll once, instead of continuing to roll as you normally would. So, we are looking at the expected gain from only one roll, and this I estimate at a minimum of 150, this being rolling a 1 and a 5.)
The highest possible score is actually 4,000, achieved when one rolls all 1's (1,000 for the first three, and an extra 1,000 for each of the next three 1's rolled). If my figures are anywhere near right, it would still be correct to roll here, since your expected loss is 1/30 of 4,000, or 133, although that gets close to the expected gain and in reality it would be hard to risk that many points on another roll.
Facebook rules require one to keep rolling until 300 points have been scored. This leads to the interesting question of whether you would ever want to re-roll a 5. This situation does come up fairly often. Say on your last roll you rolled a 5 and another counter (either 1 or 5). Your total is now 250. You can roll two dice with the 250 score, or pick up a 5 and roll three dice with a 200 score. I like the idea of rolling three dice because of the possibility of rolling three of the same number. However, I haven't yet been able to get the odds to come out in favor of re-rolling the 5. Stay tuned!