Even More Farkle Odds

A situation came up this morning which left me unsure what to do. I had scored all six dice and was rolling again on the same turn. I rolled a one, so I have a decision of rolling the remaining five dice or quitting. I think I was at 700 or so.

Naturally one has to figure the odds for future situations like this. Rolling four dice the odds of rolling no 1's or 5's is 256 out of 1296. But there are about 52 other ways to score, this being three of a kind. This makes the odds of farkling 204 out of 1296, or about 16%.

Rolling five dice, there are 1024 rolls containing no 1 or 5, but about 360 of those rolls contain at least three of a kind, so odds of farkling with five dice are 664 out of 7776, or about 8.5%.

What you should do depends on your estimate of the expected gain in your score by rolling. In the table below I have estimated an expected gain by rolling for 4, 5, and 6 dice.

# of Dice---- Farkle Odds---- Expected Gain-------- Break-even Point

---6----------- 3.3% -------------200 est.---------------6,060

---5------------8.5%...................150 est........................1,765

---4-----------15.7%-------------113 est..........................720

---3----------- 27.8%-------------86.9--------------------312.6

---2-----------44.4%--------------50---------------------112.6

---1-----------66.7%--------------25-----------------------37.5