Yahtzee is a dice game involving a combination of chance and strategy. A player begins their turn by rolling five dice. After this roll, the player may decide to re-roll any number of the dice. At most, there are a total of three rolls for each turn. Following these three rolls, the result of the dice is entered onto a score sheet. This score sheet contains different categories, such as a full house or large straight. Each of the categories is satisfied with different combinations of dice.

The most difficult category to fill-in is that of a Yahtzee. A Yahtzee occurs when a player rolls five of the same number. Just how unlikely is a Yahtzee? This is a problem that is much more complicated than finding probabilities for two or even three dice. The main reason is that there are many ways to obtain five matching dice during three rolls.

We can calculate the probability of rolling a Yahtzee by using the combinatorics formula for combinations, and by breaking the problem into several mutually exclusive cases.

## One Roll

The easiest case to consider is obtaining a Yahtzee immediately on the first roll. We will first look at the probability of rolling a particular Yahtzee of five twos, and then easily extend this to the probability of any Yahtzee.

The probability of rolling a two is 1/6, and the outcome of each die is independent of the rest. Thus the probability of rolling five twos is (1/6) x (1/6) x (1/6) x (1/6) x (1/6) = 1/7776. The probability of rolling five of a kind of any other number is also 1/7776. Since there are a total of six different numbers on a die, we multiply the above probability by 6.

This means that the probability of a Yahtzee on the first roll is 6 x 1/7776 = 1/1296 = 0.08 percent.

## Two Rolls

If we roll anything other than five of a kind of the first roll, we will have to re-roll some of our dice to try to get a Yahtzee. Suppose that our first roll has four of a kind. we would re-roll the one die that doesn’t match and then get a Yahtzee on this second roll.

The probability of rolling a total of five twos in this way is found as follows:

- On the first roll, we have four twos. Since there is a probability 1/6 of rolling a two, and 5/6 of not rolling a two, we multiply (1/6) x (1/6) x (1/6) x (1/6) x (5/6) = 5/7776.
- Any of the five dice rolled could be the non-two. We use our combination formula for C(5, 1) = 5 to count how many ways we can roll four twos and something that is not a two.
- We multiply and see that the probability of rolling exactly four twos on the first roll is 25/7776.
- On the second roll, we need to calculate the probability of rolling one two. This is 1/6. Thus the probability of rolling a Yahtzee of twos in the above way is (25/7776) x (1/6) = 25/46656.

To find the probability of rolling any Yahtzee in this way is found by multiplying the above probability by 6 because there are six different numbers on a die. This gives a probability of 6 x 25/46656 = 0.32 percent.

But this is not the only way to roll a Yahtzee with two rolls. All of the following probabilities are found in much the same way as above:

- We could roll three of a kind, and then two dice that match on our second roll. The probability of this is 6 x C(5 ,3) x (25/7776) x (1/36) = 0.54 percent.
- We could roll a matching pair, and on our second roll three dice that match. The probability of this is 6 x C(5, 2) x (100/7776) x (1/216) = 0.36 percent.
- We could roll five different dice, save one die from our first roll, then roll four dice that match on the second roll. The probability of this is (6!/7776) x (1/1296) = 0.01 percent.

The above cases are mutually exclusive. This means that to calculate the probability of rolling a Yahtzee in two rolls, we add the above probabilities together and we have is approximately 1.23 percent.

## Three Rolls

For the most complicated situation yet, we will now examine the case where we use all three of our rolls to obtain a Yahtzee. We could do this in several ways and must account for all of them.

The probabilities of these possibilities are calculated below:

- The probability of rolling four of a kind, then nothing, then matching the last die on the last roll is 6 x C(5, 4) x (5/7776) x (5/6) x (1/6) = 0.27 percent.
- The probability of rolling three of a kind, then nothing, then matching with the correct pair on the last roll is 6 x C(5, 3) x (25/7776) x (25/36) x (1/36) = 0.37 percent.
- The probability of rolling a matching pair, then nothing, then matching with the correct three of a kind on the third roll is 6 x C(5, 2) x (100/7776) x (125/216) x (1/216) = 0.21 percent.
- The probability of rolling a single die, then nothing matching this, then matching with the correct four of a kind on the third roll is (6!/7776) x (625/1296) x (1/1296) = 0.003 percent.
- The probability of rolling three of a kind, matching an additional die on the next roll, followed by matching the fifth die on the third roll is 6 x C(5, 3) x (25/7776) x C(2, 1) x (5/36) x (1/6) = 0.89 percent.
- The probability of rolling a pair, matching an additional pair on the next roll, followed by matching the fifth die on the third roll is 6 x C(5, 2) x (100/7776) x C(3, 2) x (5/216) x (1/6) = 0.89 percent.
- The probability of rolling a pair, matching an additional die on the next roll, followed by matching the last two dice on the third roll is 6 x C(5, 2) x (100/7776) x C(3, 1) x (25/216) x (1/36) = 0.74 percent.
- The probability of rolling one of a kind, another die to match it on the second roll, and then a three of a kind on the third roll is (6!/7776) x C(4, 1) x (100/1296) x (1/216) = 0.01 percent.
- The probability of rolling one of a kind, a three of a kind to match on the second roll, followed by a match on the third roll is (6!/7776) x C(4, 3) x (5/1296) x (1/6) = 0.02 percent.
- The probability of rolling one of a kind, a pair to match it on the second roll, and then another pair to match on the third roll is (6!/7776) x C(4, 2) x (25/1296) x (1/36) = 0.03 percent.

We add all of the above probabilities together to determine the probability of rolling a Yahtzee in three rolls of the dice. This probability is 3.43 percent.

## Total Probability

The probability of a Yahtzee in one roll is 0.08 percent, the probability of a Yahtzee in two rolls is 1.23 percent and the probability of a Yahtzee in three rolls is 3.43 percent. Since each of these are mutually exclusive, we add the probabilities together. This means that the probability of obtaining a Yahtzee in a given turn is approximately 4.74 percent. To put this into perspective, since 1/21 is approximately 4.74 percent, by chance alone a player should expect a Yahtzee once every 21 turns. In practice, it may take longer as an initial pair may be discarded to roll for something else, such as a straight.