There are many variations on the rules of Yahtzee, such as Triple Yahtzee.

The game is also available under many other names (Kniffel, Quintzee, Omniscore, ...).

- See Links for additional background information.
- The
*expected final score*of the Optimal Solitaire Yahtzee Player is 254.5896... with a*standard deviation*of 59.6117... (numerical approximations of theoretical values). - The
*median final score*of the Optimal Solitaire Yahtzee Player is 248, that is, in (slightly more than) 50% of the games it scores at least 248 (based on simulation of one million games). - The following table lists, for each category and some other score boxes,
the
*expected score*,*variance*,*standard deviation*(square root of the variance), and the*percentage of zero scores*(i.e. probability that the category ends with a zero score) of the Optimal Solitaire Yahtzee Player (numerical approximations of theoretical values):Category Expectation Variance Std. dev. % 0 scores *Aces*1.88 1.48 1.22 10.84 *Twos*5.28 3.99 2.00 1.80 *Threes*8.57 7.36 2.71 0.95 *Fours*12.16 10.80 3.29 0.60 *Fives*15.69 14.83 3.85 0.50 *Sixes*19.19 21.56 4.64 0.53 *Upper Section Bonus*23.84 266.04 16.31 31.88 *Three of a Kind*21.66 31.56 5.62 3.26 *Four of a Kind*13.10 122.63 11.07 36.34 *Full House*22.59 54.41 7.38 9.63 *Small Straight*29.46 15.87 3.98 1.80 *Large Straight*32.71 238.42 15.44 18.22 *Yahtzee*16.87 558.88 23.64 66.26 *Chance*22.01 6.45 2.54 0.00 *Extra Yahtzee Bonus*9.58 1161.19 34.08 91.76 *GRAND TOTAL*254.59 3553.52 59.61 0.00 *Yahtzees Rolled*0.46 0.47 0.69 63.24 *Jokers Applied*0.04 0.04 0.19 96.30 *GRAND TOTAL*of the Optimal Solitaire Yahtzee Player comes from*Large Straight*(well over 30 points), with*Small Straight*as a close second (almost 30). The largest variability is, obviously, in the contribution of the*Extra Yahtzee Bonus*(slighty more than 34).The rightmost column shows that OSYP scores 0 for

*Yahtzee*in almost 2 out of every 3 games. Or, to formulate it more positively, OSYP scores 50 for*Yahtzee*once in every 3 games. In fact, OSYP rolls, on average, 5 equals almost every other game, and obtains the*Extra Yahtzee Bonus*of 100 points almost once in every 10 games.Notice where the main contributions to the expected final score come from. Also note the size of the variance:

*Large Straight*and*Upper Section Bonus*with high variance, but*Small Straight*and*Chance*with low variance. Here they are in descending order of expected contribution:Category Expectation Std. dev. *Large Straight*32.71 15.44 *Small Straight*29.46 3.98 *Upper Section Bonus*23.84 16.31 *Full House*22.59 7.38 *Chance*22.01 2.54 *Three of a Kind*21.66 5.62 *Sixes*19.19 4.64 *Yahtzee*16.87 23.64 *Fives*15.69 3.85 *Four of a Kind*13.10 11.07 *Fours*12.16 3.29 *Extra Yahtzee Bonus*9.58 34.08 *Threes*8.57 2.71 *Twos*5.28 2.00 *Aces*1.88 1.22 The scores in the separate categories are not all independent. Even so, the expectation for the

*GRAND TOTAL*is the sum of the expected values for the constituent categories. However, the variance of the*GRAND TOTAL*(3553.52) does not equal the sum of the variances of the constituent categories (2515.47). The difference of 1038.05 can be explained by dependence.For two random variables

*X*and*Y*, we have- E[
*X*+*Y*] = E[*X*]+E[*Y*] - Var[
*X*+*Y*] = Var[*X*]+Var[*Y*] + 2*Cov[*X*,*Y*]

where Cov[

*X*,*Y*] is the*covariance*, which measures how*X*and*Y*"co-vary", that is, how they vary together. This covariance is positive if*X*and*Y*"tend" in the same direction (a high value for*X*occurs frequently with a higher value for*Y*), and it is negative if they "tend" in opposite direction (a high value for*X*occurs frequently with a lower value for*Y*). When*X*and*Y*are statistically independent, their covariance is zero. Note that Cov[*X*,*Y*] = Cov[*Y*,*X*] and Cov[*X*,*X*] = Var[*X*]. - E[
- The following table shows the
*covariance*and*correlation*matrix for all score boxes. The values on the diagonal are the*variances*, the values above the diagonal in the upper-right triangle are the*covariances*, and the values below the diagonal in the lower-left triangle are the*statistical correlations*(ranging between -1 and 1; only values outside the interval -0.05 to +0.05 shown). These are numerical approximations of theoretical values. [ text file with covariances, sorted; correlations, sorted ]Category 1 2 3 4 5 6 U T F H S L Y C E G *Aces*(1)1.48 -0.06 -0.06 -0.05 -0.03 0.01 1.94 -0.08 0.07 0.03 -0.00 0.55 -0.21 -0.05 1.32 4.86 *Twos*(2). 3.99 -0.07 -0.11 -0.07 -0.05 5.76 -0.29 -0.11 0.12 -0.09 0.50 -0.84 -0.00 2.29 10.97 *Threes*(3). . 7.36 -0.05 -0.06 -0.04 11.89 -0.43 0.19 0.09 -0.13 0.57 -0.96 0.07 3.79 22.17 *Fours*(4). . . 10.80 0.05 0.19 18.76 -0.46 0.81 0.05 -0.13 0.50 -0.45 0.21 5.57 35.71 *Fives*(5). . . . 14.83 0.40 25.80 -0.12 1.89 0.08 -0.14 0.47 -0.17 0.40 7.28 50.61 *Sixes*(6). . . . . 21.56 36.77 0.75 3.56 0.27 -0.08 0.91 0.08 0.78 10.10 75.20 *Upper Section Bonus*(U)*0.10**0.18**0.27**0.35**0.41**0.49*266.04 -1.73 12.46 0.03 -1.09 -0.11 -11.25 2.88 32.86 401.02 *Three of a Kind*(T). . . . . . . 31.56 1.13 1.37 0.08 0.60 -9.74 0.77 -6.88 16.54 *Four of a Kind*(F). . . . . *0.07**0.07*. 122.63 0.51 -0.12 9.52 -4.99 1.89 15.83 165.27 *Full House*(H). . . . . . . . . 54.41 0.19 4.81 -8.08 0.84 -0.20 54.51 *Small Straight*(S). . . . . . . . . . 15.87 -0.94 -2.90 0.11 -0.94 9.68 *Large Straight*(L). . . . . . . . *0.06*. . 238.42 -5.08 2.58 32.97 286.27 *Yahtzee*(Y). . . . . . . *-0.07**-0.05*. . . 558.88 -3.77 317.41 827.91 *Chance*(C). . . . . *0.07**0.07**0.05**0.07*. . *0.07**-0.06*6.45 -1.47 11.69 *Extra Yahtzee Bonus*(E). . . *0.05**0.06**0.06**0.06*. . . . *0.06**0.39*. 1161.19 1581.12 *GRAND TOTAL*(G)*0.07**0.09**0.14**0.18**0.22**0.27**0.41**0.05**0.25**0.12*. *0.31**0.59**0.08**0.78*3553.52 We now see that the difference of 1038.05 noted above, is explained by the dependences between

*Yahtzee*and the*Extra Yahtzee Bonus*(almost 640),- the Upper Section categories
*Aces*through*Sixes*and the*Upper Section Bonus*(about 200), and - the
*Extra Yahtzee Bonus*and other categories (about 200)

Also note

- the low covariances among Upper Section categories, many of them negative;
- the negative covariances between
*Yahtzee*and most other categories. - the total contribution of negative covariances is only about -129.

- The following table lists, for each category, the
*earliest turn*in which the Optimal Solitaire Yahtzee Player scores `non-zero' or `zero' for it:Earliest turn scoring Category Non-Zero Zero *Aces*1 2 *Twos*1 3 *Threes*1 4 *Fours*1 5 *Fives*1 6 *Sixes*1 9 *Three of a Kind*1 7 *Four of a Kind*2 2 *Full House*1 5 *Small Straight*1 10 *Large Straight*1 7 *Yahtzee*1 3 *Chance*1 never

Turn | Final Roll | Score | Category |
---|---|---|---|

1 | 1 4 4 5 5 | 1 | Aces |

2 | 1 2 3 5 5 | 2 | Twos |

3 | 1 1 2 2 6 | 0 | Four of a Kind |

4 | 1 2 2 4 6 | 0 | Yahtzee |

5 | 1 1 2 2 6 | 0 | Threes |

6 | 1 2 2 3 3 | 0 | Fours |

7 | 1 2 2 3 3 | 0 | Fives |

8 | 1 2 2 3 3 | 0 | Full House |

9 | 1 2 2 3 3 | 0 | Sixes |

10 | 1 1 2 3 3 | 0 | Large Straight |

11 | 1 1 2 2 3 | 9 | Chance |

12 | 4 5 5 6 6 | 0 | Three of a Kind |

13 | 5 6 6 6 6 | 0 | Small Straight |

12 | GRAND TOTAL |

If you are paranoid, then you can guarantee a minimum score of 18. (But you will not do well on average!)

© 1999, Tom Verhoeff (TUE, Math/CS)

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