Yahtzee is a dice game involving a combination of chance and strategy. On a player’s turn, he or she begins by rolling five dice. After this roll, a player may decide to reroll any number of 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 for this is that there are a number of 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%.

### Two Rolls

If we roll anything other than five of a kind of the first roll, we will have to reroll some of our dice to try to get a Yahtzee. Suppose that our first roll has four of a kind, we reroll 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%

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%.
- 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 %
- 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%.

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%.

### 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 different ways and must account for all of them.

The probabilities 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%.
- 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%.
- 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%.
- 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%
- 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%.
- 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%.
- 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%.
- 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%.
- 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%.
- 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%.

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%.

### Total Probability

The probability of a Yahtzee in one roll is 0.08%, the probability of a Yahtzee in two rolls is 1.23% and the probability of a Yahtzee in three rolls is 3.43%. 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%. To put this into perspective, since 1/21 is approximately 4.74%, 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 in order to roll for something else, such as a straight.