Optimal Solitaire Yahtzee Player: Trivia

Yahtzee is a registered trademark of Milton Bradley, which is owned by Hasbro, Inc..

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, ...).

Characteristics of the Optimal Solitaire Yahtzee Player

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

The single most valuable contribution to the expected 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].

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 diffence 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 betweenYahtzee 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

Noteworthy is that the Optimal Solitaire Yahtzee Player never opts for Four of a Kind in its first turn, but any other category is a candidate (if circumstances demand so). It is also interesting to see that OSYP delays zeroing out for Small Straight until at least the 10th turn. Aces and Four of a Kind, however, can be sacrificed as early as the 2nd turn, and Yahtzee as early as the 3rd turn.

Under worst case conditions (extremely bad luck with the dice), the Optimal Solitaire Yahtzee Player obtains a minimum score of 12.

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!)