# Yahtzee Odds and Probabilities

The Yahtzee scorecard consists of 13 scoring categories, with the possibility for multiple Bonus Yahtzees. It is theoretically possible to achieve an ultimate high score of 1535. A player would achieve this feat by rolling a Yahtzee on each turn and using the ensuing Jokers to complete the Lower Section categories. Many of the Yahtzees would need to be in a certain number: the Upper Section's Ones in ones, Twos in twos, etc., and in sixes to max out Three of a Kind, Four of a Kind, and Chance in the Lower Section. The odds of approaching anything near a 1535 grand total are astronomical. The average Yahtzee player generally considers a final score in the 250 point range to be a success, and indeed this turns out to be about the average score for the competent player.

Researchers at the World Yahtzee Institute have analyzed thousands of games to determine the odds of achieving a particular result:

Grand Total | Probability |
---|---|

>150 | 98% |

>200 | 78% |

>250 | 59% |

>300 | 16% |

>400 | 4% |

>500 | 1% |

The following graph illustrates the odds of rolling a particular combination on the first, second, and third roll of a turn. The Full House and Straights can not be calculated generally because of the multitude of variations that exist for each roll. This would need to be calculated on a case-by-case basis depending on each of the individual player's rolls. The figures assume that the player goes for the given scoring category from the first turn and does not deviate in their strategy. Otherwise it may take longer to achieve a given score as an initial pair may be discarded to roll for something else.

Three of a Kind | Four of a Kind | Full House | Small Straight | Large Straight | Yahtzee | |
---|---|---|---|---|---|---|

Roll 1 | 19.29% | 1.93% | 3.86% | 12.35% | 3.09% | 0.08% |

Roll 2 | 40.90% | 11.97% | - | - | - | 1.26% |

Roll 3 | 45.24% | 24.48% | - | - | - | 4.6% |

Take a deep-dive into the mathematics behind Yahtzee odds and probabilities at these sites: