Farkle Probabilities

Number of dice
thrown		 Probability to
Farkle		 Probability to
NOT Farkle		 Expectation Value
of throw		 Stratagy
one 2/3 = 66.7% 1/3 = 33.3% 25 points Never throw only
one dice
two (2/3)2=4/9 ≈ 44% 5/9 ≈ 55% 61.1 points Don't throw only
two dice
three 15/54 ≈ 28% 39/54 ≈ 72% 86.8 points Throw 3 dice if there
are less than 223 points
on the line
four ∼ 15.7% ∼ 84.3% 143.5 points Usual rule: Throw 4 dice if
there are less than 300 points
after throwing 5 dice. See below
for exception.
five ∼ 7.7% ∼ 92.3% 225.8 points Usual rule: Throw 5 dice if
there are less than 300 points
after throwing 6 dice. See below.

Expectation Value

The expectation value is the average value one expects when throwing N dice. For example, for N=1 one can get 50 points if a "5" is thrown, 100 points if a "1" is thrown and 0 points if a "2", "3", "4", or "6" are thrown. Each outcome has a 1/6 chance of occuring. So the expectation value for E1 (i.e. one dice) is

E1 = (1/6)50 + (1/6)100 + (4/6)0 = 25 points.

Strategy

For throwing 3 dice:
Suppose that there are "x" points on the line. For example if you have two fives and one "1", you have 200 points on the line. To decide if one should throw the remaining three dice or not, the reasoning goes as follows: If you throw, then you have a probability of 72% of getting (x+E3) = (x+86.8) points. But you have a probability of 28% of losing all x points. So, you should throw the three dice if

(x+86.8)(0.72) > x
86.8(0.72) > x (1-0.86)
86.8(0.72/0.28) > x
or x<223

Thus, if there are less then 223 points on the line, then the best strategy is to throw the remaining three dice.

For throwing 4 dice:
For 4 dice, the "decision making" equation is different than with 3 dice. On the line, one can have two 5's (100 points), two 1's (200 points) or a one and a five (150 points). Let this number be y. Then for the three possible values of y and the expected value after throwing 4 dice, we have

y=100: (100+143.5)(0.843) = 205
y=150: (150+143.5)(0.843) = 247
y=200: (200+143.5)(0.843) = 289

Thus, after throwing 5 dice, if there are less than 250 points and y<200, then put a one or five on the line and throw the remaining 4 dice. However, if after throwing the 5 dice there are 300 or more points, then stop.

For throwing 5 dice:
For 5 dice, the "decision making" equation is similar to the 4 dice case. There are two possibilities for the value On the line: 50 points (one five) or 100 points (one one). Let this number be y. The two possibilities and their expectation values are

y=50: (50+225.8)(0.923) = 254.6
y=100: (100+225.8)(0.923) = 300.7

Thus, if there are only one or two "5"'s after the initial throw and the point total is less than 250, put one "5" on the line and throw the remaining 5 dice. If there are one or two "1"'s after the first throw and the point total is less than 300, then put one "1" on the line and throw the remaining 5 dice. If there are three "2"'s and one "5" for a total of 250 points after the initial roll, then there is only a small advantage to putting the "5" on the line and throwing 5 dice. If there are three "2"'s and one "1" for a total of 300 points after the initial roll, then there is no real advantage to putting the "1" on the line and throwing 5 dice.