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Theoretical Probability of Rolling Certain Scores

It seems very attractive to try and calculate the probability of completing various target scores (such as full house) from various starting positions, but in fact its appalling hard to do because there are so many combinations possible. The first roll is entirely random, but has dramatic affect on the subsequent chance of achieving a given roll, so there's no fixed starting point for an analysis. Also, the chances are affected by what has already been rolled. If you've used 5's and three of a kind already, a start throw of 5-5-3-6-4 is much less likely to yield a good result than it would at the start of a game.

The chance of rolling a yahtzee in one turn been calculated* as 4.6% or about 1 in 22 rolls but even that assumes that the player is targeting a yatzee and doesn't get tempted to go for a straight after the first roll, or take a random full house after roll two. These are things that happen in real games, so the real odds are a little less.
And this is just an analysis on one roll type, not taking into account previous scores or choices of how to select a given roll.

The attempt to statistically assess the likelihood of acheiving a given total game score is imponderably difficult .

* References - Tim Fulford   Cornell University  if you want to look at the derivation for yourself. In Feburary 2008 Tim went on to produce this simplest yet version of the maths.


Measured Probability of Achieving Certain Scores

In December 2007 Chris Young, a very experienced Yahtzee player, played 300 games of Yahtzee using an electronic hand held device  (using the electronic device has the useful effect of gauranteeing the yahtzee rules were ruthlessly followed). The following stats were found

Number of Yahtzee's rolled = 105, or one every 2.85 games.

Low Score - 127

Highest Score with no yahtzee - 277

Highest Score with 1, 2 and 3 yahtzees - 326, 437, 504.

The most interesting thing to come from this trial was the probability of achieving and overall game score, shown by this table and graph

Score At Least Probability
150 98%
200 81
250 45%
300 13%
400 3%
500 1%

graph of yahtzee score probabilities


Computer Generated Scores

In January 2008 Tim Fulford and Oliver Humpidge independantly wrote alogrithms to simulate Yahtzee games. Oliver ran his script for 300,000 turns and found about 1:22 resulted in Yahtzee which matches theory very closely. Tim ran his for an impressive 15 million and came up with 1:27, but his algorithm didn't seem so good at playing the game, because it got a low score of just 37 which its doubtful if a  human player could do without deliberately aiming for a low score. Its high score was 918. It also showed only 60% chance of scoring at least 200, compared to over 80% by human expert Chris Young. Of course its high score is better due to the sheer number of games played.

Run your own computer simulation

You can download this programme by Aaron Murgatroyd and run as many simulated games as you like, watching the developement of high, low and average scores over time.