Chapter 12.1, Problem 32E

(a)

To determine

To Explain: the reason of wrong of researcher.

Explanation of Solution

Given:

Suppose

A= rolling the dice with 1 die

B= rolling the 1 with all 5 dice

C= rolling a yathzee

Formula used:

Multiplication rule:

  $P\left(A\cap B\right)=P\left(AandB\right)=P\left(A\right)\times P\left(B\right)$

Addition rule:

  $P\left(A\cup B\right)=P\left(AorB\right)=P\left(A\right)+P\left(B\right)$

Calculation:

Dice are fair and since the dice is having total 6 possible outcomes so to roll a 1.

  $P\left(A\right)=\frac{1}{6}$

Using the multiplication rule

  $\begin{array}{l}P\left(B\right)=P\left(A\right)\times P\left(A\right)\times \mathrm{......}\times P\left(A\right)\left\{5timerepetitions\right\}\\ ={\left[P\left(A\right)\right]}^5\\ ={\left[\frac{1}{6}\right]}^5\end{array}$

Although rolling a yathzee, roll the same number along with all 5 dice. There are total 6 possible numbers that could roll (, 2, 3, 4, 5, 6) and each of these numbers have the same probability of being rolled with all 5 dice.

Using the addition rule for the event of mutually exclusive

  $P\left(C\right)=6P\left(B\right)=6{\left(\frac{1}{6}\right)}^5$

(b)

To determine

To Calculate: the appropriate probability to help the answer.

Answer to Problem 32E

No

Explanation of Solution

Given:

Suppose

A= rolling the dice with 1 die

B= rolling the 1 with all 5 dice

C= rolling a Yathzee

Formula used:

Multiplication rule:

  $P\left(A\cap B\right)=P\left(AandB\right)=P\left(A\right)\times P\left(B\right)$

Addition rule:

  $P\left(A\cup B\right)=P\left(AorB\right)=P\left(A\right)+P\left(B\right)$

Calculation:

Dice are fair and since the dice is having total 6 possible outcomes so to roll a 1.

  $P\left(A\right)=\frac{1}{6}$

Using the multiplication rule

  $\begin{array}{l}P\left(B\right)=P\left(A\right)\times P\left(A\right)\times \mathrm{......}\times P\left(A\right)\left\{5timerepetitions\right\}\\ ={\left[P\left(A\right)\right]}^5\\ ={\left[\frac{1}{6}\right]}^5\end{array}$

Although rolling a yathzee, roll the same number along with all 5 dice. There are total 6 possible numbers that could roll (, 2, 3, 4, 5, 6) and each of these numbers have the same probability of being rolled with all 5 dice.

Using the addition rule for the event of mutually exclusive

  $P\left(C\right)=6P\left(B\right)=6{\left(\frac{1}{6}\right)}^5=\frac{1}{1296}$

Using the addition rule and complement rule for the mutually exclusive

  $\begin{array}{l}P\left(X\le 25\right)=P\left(X=1\right)+P\left(X=2\right)+\mathrm{...}P\left(X=24\right)+P\left(X=25\right)\\ ={\displaystyle \sum _{k=1}^{25}{\left(1-\frac{1}{1296}\right)}^{k-1}\left(\frac{1}{1296}\right)}\\ =0.0191\end{array}$

It is observed that the probability is small, that is indicates that if is very not likely to get a Yathzee in the first 25 rolls and therefore Nassir should not be surprised that he has not gotten a Yathzee after 25 rolls.