Chapter 10.1, Problem 36E

Solution

To calculate: The probability of rolling five of a kind (which is Yahtzee!) on the first roll.

The probability of rolling five of a kind on the first roll is $\frac{1}{2296}$ .

Given information:

On the first roll in a game of Yahtzee, five dice are tossed simultaneously.

Calculation:

First find the total number of outcomes. Since there are six numbers on a die, raise this to the $5\text{th}$ power since there are $5$ dice.

  $\begin{array}{l}{6}^5=6\times 6\times 6\times 6\times 6\\ =7776\end{array}$

Since there are $6$ ways to roll a $5$ of a kind (five $1\text{'s}$ , five $\text{2's}$ , five $\text{3's}$ , five $4\text{'s}$ , five $5\text{'s}$ , and five $6\text{'s}$ ), the probability is $6$ out of $7776$ . This can then be simplified to $1$ out of $1296$ :

  $\frac{6}{7776}=\frac{1}{1296}$

Hence, probability of rolling five of a kind (which is Yahtzee!) on the first roll is $\frac{1}{2296}$ .