Yahtzee! Part 1

Probability Level 3

In the game of Yahtzee, each player's turn consists of rolling five 6-sided dice. The highest possible score is obtained when a player rolls all five dice as the same number. Whenever this happens, this is called a "Yahtzee" (hence the name of the game).

During each player's turn, that player may re-roll the dice up to two more times after the first roll. However, we will ignore this rule for this problem.

If the probability of rolling a Yahtzee in the first roll is $\frac{a}{b}$ , where $a$ and $b$ are positive co-prime integers, then what is $a+b$ ?

The answer is 1297.

1 solution

Andy Hayes
Dec 6, 2015

We can consider each single dice roll to be an independent event. Therefore, in order to obtain the probability of an event for all five dice, we multiply probabilities for each individual dice.

Let's first consider the probability of rolling a Yahtzee consisting of all 1s. The probability of rolling a 1 on a single dice is $\frac{1}{6}$ . Therefore, the probability of rolling a 1 on all five dice would be: $\large\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}=\frac{1}{6^5}$ .

Now consider the probability of rolling a Yahtzee consisting of all 2s. The calculation is the exact same, and so the probability of rolling a 2 on all five dice would be $\large\frac{1}{6^5}$ .

There are six possible Yahtzees corresponding to each of the numbers on the dice, and the probability of rolling each Yahtzee is the same $\large\frac{1}{6^5}$ . In addition, each possible Yahtzee is mutually exclusive. Therefore, the probability of rolling any Yahtzee would be $\large6\times\frac{1}{6^5}=\frac{1}{6^4}=\frac{1}{1296}$ .

Thus, $a+b=\boxed{1297}$ .

Yahtzee! Part 1

The answer is 1297.

1 solution

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