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, ...).
The exact expected final score of the Optimal Solitaire Yahtzee Player is the fraction with numerator
4837423778178559905031968925883862839950072036299117171073109676394093468257472452669937843237871315151422338846646267957265652473530478355and denominator
19000886425012870870451396938693991151248393833957109655563098372206519682663244561350767261332742248200665160873051839214724243263586304This number was determined on 12-Jun-2017 by Jeffrey Liese and Katy Kelly using Mathematica 11, running for about 5 days on a PC with an Intel Core i7-4770K CPU @ 3.5 GHz with 16 GB RAM. [Publication to appear]
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 |
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
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].
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
Also note
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!)