Large Straight or Yahtzee that is thrown. The best strategy is also to keep a four-of-a-kind and try to throw a Yahtzee, even after the first throw. The player should not keep a three-of-a-kind. Keep a small straight or 4 out of 5 (e.g. 1235). Otherwise one should simply keep any 2, 3, 4, or 5 (just one of each) so that with 12256 one would keep 25. On average one will succeed 26.53% of the time and score an average of 10.61. Yahtzee A player should keep any Yahtzee, four-of-a-kind, three-of-a-kind, or pair that is thrown and re-roll the others. On average one will succeed 4.60% of the time and score an average of 2.30. Chance To get the maximum average score the strategy is straightforward. After the first throw, the player will keep any 5s and 6s. After the second throw one will keep any 4s, 5s, and 6s. On average one will score 23.33. Different strategies will be required when one needs to achieve a specific target. Optimal strategy The strategy for maximising the expected score has been determined. The average score when using this strategy is 254.59. This average is based on calculations using the "free choice Joker rule". Optimal strategy using the "forced Joker rule" has also been explored and the average score when using this strategy has been determined exactly as a rational number. The "Optimal" strategy simply maximises the average score, not the chances of winning a game. There are two main reasons for this. Firstly, the Optimal strategy takes no account of any opponents. In normal gameplay a player will adjust their strategy depending on the scores of the other player or players. Secondly, the Optimal strategy tends to give undue importance to Yahtzee bonuses. It is rare for a player without a Yahtzee bonus to beat a player with one. So, in normal gameplay, a player's strategy is not significantly influenced by the value given for a Yahtzee bonus. Consider the situation where the Yahtzee bonus was worth a million rather than a hundred. It would not influence normal gameplay where the objective is to score more than the opponent. It would, however, affect the "Optimal" strategy, though, since scoring a million would have a dramatic effect on the average score. The "Optimal" strategy would be dominated by the prospect of a Yahtzee bonus. Even with a Yahtzee bonus worth 100