Chapter PV, Problem 8RE

To determine

To find out what is the approximate probability that your total is between $\text{30 and 40}$ and specify the model you use and the assumptions and conditions that justify your approach.

Answer to Problem 8RE

The approximate probability that your total is between $\text{30 and 40}$ is $64.76\%$ .

Explanation of Solution

It is given in the question that when one die is rolled, the number of spots showing has a mean of $\text{3}\text{.5}$ and a standard deviation of $\text{1}\text{.7}$ and suppose you roll ten dice. Thus, the assumptions and conditions for this model is as:

Random condition: It is satisfied as the dice rolled are random.

Independence condition: It is satisfied as the dice that are rolled are independent.

  $10\%$ condition: It is satisfied as the dice rolled is $10\%$ less than of all dice rolled.

Thus, all the conditions and assumptions are met. So, the mean and the standard deviation is calculated as:

  $\begin{array}{l}\text{Mean}=np\\ =3.5\times 10\\ =35\\ \text{Standard deviation}=\sqrt{npq}\\ =\sqrt{10\times {1.7}^2}\\ =5.376\end{array}$

Now, we will calculate the z -score as:

  $\begin{array}{l}z=\frac{30-35}{5.376}\\ =-0.93\\ z=\frac{40-35}{5.376}\\ =0.93\end{array}$

So, we will calculate by the calculator as:

  $\begin{array}{l}normalcdf(-0.93,0.93)=0.6476\\ =64.76\%\end{array}$

Thus, the approximate probability that your total is between $\text{30 and 40}$ is $64.76\%$ .