Probability of a Small 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, the probability can be used to analyze how likely it is to roll a small 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 small 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 small straight is the number of ways to roll a small 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 the fractions that we use for our probability.

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

By Courtney Taylor

Number of Straights

Next, we need to know how many ways there are to roll a small straight. This is more difficult than calculating the size of the sample space. We begin by counting how many straights are possible.

A small straight is easier to roll than a large straight, however, it is harder to count the number of ways of rolling this type of straight. A small straight consists of exactly four sequential numbers. Since there are six different faces of the die, there are three possible small straights: {1, 2, 3, 4}, {2, 3, 4, 5} and {3, 4, 5, 6}. The difficulty arises in considering what happens with the fifth die. In each of these cases, the fifth die must be a number that does not create a large straight. For example, if the first four dice were 1, 2, 3, and 4, the fifth die could be anything other than 5. If the fifth die was a 5, then we would have a large straight rather than a small straight.

This means that there are five possible rolls that give the small straight {1, 2, 3, 4}, five possible rolls that give the small straight {3, 4, 5, 6} and four possible rolls that give the small straight {2, 3, 4, 5}. This last case is different because rolling a 1 or a 6 for the fifth die will change {2, 3, 4, 5} into a large straight. This means that there are 14 different ways that five dice can give us a small straight.

Now we determine the different number of ways to roll a particular set of dice that give us a straight. Since we only need to know how many ways there are to do this, we can use some basic counting techniques.

Of the 14 distinct ways to obtain small straights, only two of these {1,2,3,4,6} and {1,3,4,5,6} are sets with distinct elements. There are 5! = 120 ways to roll each for a total of 2 x 5! = 240 small straights.

The other 12 ways to have a small straight are technically multisets as they all contain a repeated element. For one particular multiset, such as [1,1,2,3,4], we will count the number od different ways to roll this. Think of the dice as five positions in a row:

• There are C(5,2) = 10 ways to position the two repeated elements among the five dice.
• There are 3! = 6 ways to arrange the three distinct elements.

By the multiplication principle, there are 6 x 10 = 60 different ways to roll the dice 1,1,2,3,4 in a single roll.

There are 60 ways to roll one such small straight with this particular fifth die. Since there are 12 multisets giving a different listing of five dice, there are 60 x 12 = 720 ways to roll a small straight in which two dice match.

In total there are 2 x 5! + 12 x 60 = 960 ways to roll a small straight.

Probability

Now the probability of rolling a small straight is a simple division calculation. Since there are 960 different ways to roll a small straight in a single roll and there are 7776 rolls of five dice possible, the probability of rolling a small straight is 960/7776, which is close to 1/8 and 12.3%.

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 small 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.