Q.26 - The Practice of Statistics for AP

Question

Yahtzee $(5.3, 6.3)$ In the game of Yahtzee, $5$ six-sided dice are rolled simultaneously. To get a Yahtzee, the player must get the same number on all $5$ dice. 

(a) Luis says that the probability of getting a Yahtzee in one roll of the dice is ${\left(\frac{1}{6}\right)}^5$. Explain why Luis is wrong.

(b) Nassir decides to keep rolling all 5 dice until he gets a Yahtzee. He is surprised when he still hasn’t gotten a Yahtzee after $25$ rolls. Should he be? Calculate an appropriate probability to support your answer .

Step-by-Step Solution

Answer

a). The probability of getting a Yahtzee is :

$\left(\text{ Yahtzee }\right)=6\times {\left(\frac{1}{6}\right)}^5$

b). The probability is $0.9808$.

Part (a) Step 1: Given Information

$5$ six sides dice have been rolled simultaneously.

To obtain Yahtzee, all $5$ dice must have same numbers.

According to Luis, the probability of getting a Yahtzee is ${\left(\frac{1}{6}\right)}^5$.

Part (a) Step 2: Explanation

The following numbers could be obtained by players $1,2,3,4,5$ and $6$.

Here all the dice are independent.

Thus, the probability of getting a Yahtzee is calculated as:

$P\left(\text{ Yahtzee }\right)=6\times {\left(\frac{1}{6}\right)}^5$

Part (b) Step 1: Given Information

$5$ six sides dice have been rolled simultaneously.

To obtain Yahtzee, all $5$ dice must have same numbers.

According to Luis, the probability of getting a Yahtzee is ${\left(\frac{1}{6}\right)}^5$.

Part (b) Step 2: Explanation

From above part, it is known that

$$\begin{gathered} P\left(\text{ Yahtzee }\right)=6\times {\left(\frac{1}{6}\right)}^5 \\ =0.0007716 \end{gathered}$$

$$\begin{gathered} P\left(\text{ No Yahtzee }\right)=1-6\times {\left(\frac{1}{6}\right)}^5 \\ =0.9992284 \end{gathered}$$

The probability of not getting Yahtzee even after $25$ rolls is computed as:

$$\begin{gathered} P(\text{ No Yahtzee }{)}^{25}=(0.9992254{)}^{25} \\ =0.9809 \\ =98.09\% \end{gathered}$$

Since, the probability is above $0.05$. Thus, it could be concluded that the probability of not getting Yahtzee is likely. Thus, Nasir should not be surprised.