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
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
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
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