3 Apr

# Yahtzee Scoring Expectations and Dream Games

Evaluation of the average score in a Yahtzee game, compared to the best case scenario we’ll probably never see in our lifetime.

There’s nothing quite like a game of Yahtzee. It cannot be compared to any card game. In Euchre, Hearts, Spades, Bridge, etc., the hand you are dealt determines your chance of winning. In Yahtzee, every players’ odds change drastically every single time they touch the dice or mark a score. Skill and scientific knowledge may get you further, but it truly is a game of fate and fortune.

Speaking of scientific data, yesterday we examined the mathematical probabilities of rolling a Yahtzee or any other desirable outcome. Today, we’ll expand on that topic by evaluating average scoring expectations.

Unfortunately, unlike so many modern day games of immense popularity, this is not something a computer can truly evaluate for us. There are so many ways a single dice roll can be judged; so many combinations to roll for; so many variables to calculate. How do you tell a computer what to roll for first, or how to judge the significance of acquiring the Upper Section Bonus compared to rolling a Yahtzee, at each and every junction in the game? All of these decisions are turn- and situation-based. It is the reason Yahtzee is such a beautiful conundrum for the scientifically minded.

Of course, such computer algorithms do exist, and have been used to evaluate scoring probabilities for years. But to get a real player’s average Yahtzee score, it takes – what else? – a real player. For this, we thank one expert Yahtzee strategist with perhaps a bit too much time on his hands. His name is Chris Young, and in December 2007, he spent no telling how long playing and recording the results of 300 games of Yahtzee on an official Yahtzee hand-held electronic game.

#### Yahtzee Scoring Probabilities by an Expert Player

As noted in one analysis of his data, the use of an electronic game eliminates human error by guaranteeing the rules of Yahtzee are “ruthlessly followed”. In Mr. Young’s 300-game evaluation, he came up with the following results:

• Yahtzee’s Rolled in 300 Games: 105 (1 in 2.85 games)
• Lowest Score in 300 (expertly played) Yahtzee Games: 127
• Highest Score in 300 (expertly played) Yahtzee Games…
• without Rolling a Yahtzee: 277
• with 1 Yahtzee rolled: 326
• with 2 Yahtzees rolled: 437
• with 3 Yahtzees rolled: 504
• Probability of scoring at least…
• 150 points = 98%
• 200 points = 81%
• 250 points = 45%
• 300 points = 13%
• 400 points = 3%
• 500 points = 1%

I believe the scoring data is the most significant here. Although Chris’s exact score average was not provided, we can estimate that the average (50%) was somewhere around 240-245 points. Also, the fact that he rolled 1 Yahtzee in every 2.85 games is significant. His results are quite a bit higher than the actual scientific probability of 4.74% of all rolls, or about 1 in every 21 rolls.

We can chalk this up to variance, being that he only performed 300 games. The standard calculation by a computer would require the conduction of one million trials to ensure probabilities are as close as possible to scientific approximation.

#### Yahtzee Scoring Probabilities by Computer Algorithms

The following month, a pair of computer programmers each developed a computerized Yahtzee algorithms to test probabilities of scoring. Tim Fulford ran his program for 300,000 trials. His partner, Oliver Humpidge, ran his program for 15 million trials. The results were quite interesting…

Fulford Algorithm: Tim’s technological assessment came up with an average of 1 Yahtzee in every 22 rolls. This is far closer to the mathematical probability of 1 in 21.1 rolls. Then again, this algorithm was written with the intent to roll Yahtzees; not necessarily to win the game.

Humpidge Algorithm: Oliver’s program, compiling data over 15 million simulations, is more far-reaching. In his assessment, a Yahtzee only rolled once in every 27 rolls; much farther from actual probabilities than you would expect for a computer program. It delivered a lower-than-average Yahtzee score with a range of 37 to 918, and only a 60% chance of finishing with a score of 200+.

The fact that Young’s expert-player assessment came up with an 81% rate of 200+ point games proves without a doubt that a good Yahtzee strategy greatly increases a players odds of scoring higher. It also proves that computer’s cannot intentionally invoke proper strategy the way an expert player would. No human could possibly score as low as 37 points unless it was their intent to score as low as possible.

#### Determining the Average Yahtzee Score

Combining all this data, with emphasis on the average scoring rate of an expert player who is clearly trying to win more so than roll Yahtzees, we find that the average Yahtzee score comes in just under 250; estimated to fall somewhere between 240-245.

However, we must take into account that the average Yahtzee is rolled about once in every other game. With this in mind, we can judge the average score, without a Yahtzee, to be somewhere in the range of 215 to 220. The average score with a Yahtzee rolled is going to be between 265-270.

#### What’s the Best Yahtzee Score Possible?

As promised, we’ll take a moment to look at the best possible game of Yahtzee one could hope for. You’d have to roll some pretty amazing dice to do it, but the highest possible score in a game of Yahtzee would be 1575. To score it requires the rolling of 13 Yahtzees in a single game, and playing by the Yahtzee Bonus Joker rule, 13-rolls-only rule, and with no other limits on Yahtzee Bonus scoring.

As such, you would score the highest possible score in all sections by rolling a Yahtzee in 1s, 2s, 3s, 4s, 5s, and 6s on top (total 140 with Upper bonus), a Yahtzee in 6s for the 3 of a Kind, 4 of a Kind and Chance, and any Yahtzee for the Full House, Small Straight, Large Straight and, of course, the actual Yahtzee. This would give you max score in every column, plus 12 Yahtzee Bonus scores worth 100 each.

The score pad would look like this…

The highest possible Yahtzee score without rolling a Yahtzee would be 301. It would require rolling a lot of 6-6-6-6-5 combinations. Although, if one were to intentionally scratch Yahtzee on the first roll, you could get a bit higher. Assuming sabotage is not the goal, the highest score without a Yahtzee would look like this…

### Worst Possible Yahtzee Score?

And of course, if we’re going to measure the best games, we might as well take a glimpse over to the opposite side of the tracks. The worst possible Yahtzee score you can get would be 5. This would require a bit of luck, paired with intentionally consistent sabotage on the players part. The end result would be a Yahtzee in 1s taken on Chance, and every other position marked with a big fat 0. It would look like this…

There’s so much more to learn about the game of Yahtzee…

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