The Probability of a Large Straight in Yahtzee in a Single Roll

Yahtzee is a dice game that uses five standard six-sided dice. On each turn, players are given three rolls to obtain several different objectives. After each roll, a player may decide which of the dice (if any) are to be retained and which are to be rerolled. The objectives include a variety of different kinds of combinations, many of which are taken from poker. Every different kind of combination is worth a different amount of points.

Two of the types of combinations that players must roll are called straights: a small straight and a large straight. Like poker straights, these combinations consist of sequential dice. Small straights employ four of the five dice and large straights use all five dice. Due to the randomness of the rolling of dice, probability can be used to analyze how likely it is to roll a large straight in a single roll.

Assumptions

We assume that the dice used are fair and independent of one another. Thus there is a uniform sample space consisting of all possible rolls of the five dice. Although Yahtzee allows three rolls, for simplicity we will only consider the case that we obtain a large straight in a single roll.

Sample Space

Since we are working with a uniform sample space, the calculation of our probability becomes a calculation of a couple of counting problems. The probability of a straight is the number of ways to roll a straight, divided by the number of outcomes in the sample space.

It is very easy to count the number of outcomes in the sample space. We are rolling five dice and each of these dice can have one of six different outcomes. A basic application of the multiplication principle tells us that the sample space has 6 x 6 x 6 x 6 x 6 = 6⁵ = 7776 outcomes. This number will be the denominator of all of the fractions that we use for our probabilities.

Number of Straights

Next, we need to know how many ways there are to roll a large straight. This is more difficult than calculating the size of the sample space. The reason why this is harder is because there is more subtlety in how we count.

A large straight is harder to roll than a small straight, but it is easier to count the number of ways of rolling a large straight than the number of ways of rolling a small straight. This type of straight consists of five sequential numbers. Since there are only six different numbers on the dice, there are only two possible large straights: {1, 2, 3, 4, 5} and {2, 3, 4, 5, 6}.

Now we determine the different number of ways to roll a particular set of dice that give us a straight. For a large straight with the dice {1, 2, 3, 4, 5} we can have the dice in any order. So the following are different ways of rolling the same straight:

• 1, 2, 3, 4, 5
• 5, 4, 3, 2, 1
• 1, 3, 5, 2, 4

It would be tedious to list all of the possible ways to get a 1, 2, 3, 4 and 5. Since we only need to know how many ways there are to do this, we can use some basic counting techniques. We note that all that we are doing is permuting the five dice. There are 5! = 120 ways of doing this. Since there are two combinations of dice to make a large straight and 120 ways to roll each of these, there are 2 x 120 = 240 ways to roll a large straight.

Probability

Now the probability of rolling a large straight is a simple division calculation. Since there are 240 ways to roll a large straight in a single roll and there are 7776 rolls of five dice possible, the probability of rolling a large straight is 240/7776, which is close to 1/32 and 3.1%.

Of course, it is more likely than not that the first roll is not a straight. If this is the case, then we are allowed two more rolls making a straight much more likely. The probability of this is much more complicated to determine because of all of the possible situations that would need to be considered.