To better understand how a perfect Yahtzee player performs, there was actually a time when a computer was encoded to play 100,000 games using the optimal strategy. Basically the following information was determined:

The high values for both the small and large straights are expected and worth about 30 points per game on average. This has to do with their high frequency of occurrence and their high scoring values of 30 and 40 points respectively. In fact the small straight only has about a 2% chance of being filled on your score card as a 0. The Yahtzee category, despite being relatively rare, has a value of about 25 points per game basically due to its high scoring value for points for the first Yahtzee and the additional 100 bonus points you get for every Yahtzee after the first one.

The computer also noted that the first six entries, Aces through Sixes all have the same averages, even though there is a slight chance of getting more Fives and Sixes, each with an average score in excess of three times the number, compared to the Aces and Twos. The Aces and Twos each have an average score of less than three times the number.

Besides Yahtzee, the category most likely to get a zero marked down for it is the four of a kind. This is due primarily to its low frequency of occurrence. In fact, in approximately one in every three games you will most likely have a zero in the score box. Of course if the dice always did what you wanted you would never have to fill any score box with zero. However, that is highly unlikely as well. Even with flawless strategy it is sometimes unavoidable as well as inevitable that you’ll eventually end up playing with dice that simply do not want to cooperate.

It should be no surprise to anyone that Yahtzee is indeed the hardest hand to make. Also not surprising, especially to those who have played the game before, is that there is a high frequency of assigning a zero to the Large Straight box as well. Also interesting to note that in roughly 10% of the games you play, the Aces box is going to be filled with a zero as well.

As a game progresses from the first hand to the 13th hand, the expected value of all the score boxes decreases. The more hands you roll the more that number decreases. What this means is that the longer a particular box remains unfilled, the lower its average score is likely to be. The rate at which the expected scores of the entries fall is closely related to the difficulty in achieving them. Therefore, the Large Straight and Yahtzee entries, with their consistent drop offs in averages, are somewhat difficult to hit in any point in the game.

The highest possible score in the game of Yahtzee is 1,575 which would require making 13 straight Yahtzees. The lowest possible score is 5 which can only be achieved by purposely filling in every box with zero except the Chance box where you are required to put in the total of your dice which has a lowest possible value of 5.