Chapter 13, Problem 13.37E

a.

To determine

Whether the events “getting a thick crust pizza” and “getting a pizza with mushrooms’ are independent or not.

Answer to Problem 13.37E

The events “getting a thick crust pizza” and “getting a pizza with mushrooms’ are not independent.

Explanation of Solution

Given info:

Thomas’s pizza shop has seven pizzas in the oven, three out of seven are thick crust, and of these only one pizza has sausage and other two have mushrooms; remaining four pizzas have regular crust, and of these two pizza has only sausage and other two has only mushrooms.

Calculation:

Let, A be the event for thick crust pizza and B be the event for mushroom pizza. Total number of pizzas is 7.

The formula for events to be independent,

$P\left(B|A\right)=P\left(B\right)$ (1)

The formula for conditional probability is,

$P\left(B|A\right)=\frac{P\left(B\text{and}A\right)}{P\left(A\right)}$ (2)

The probability for getting ‘mushroom and thick crust pizza’ is,

$P\left(B\text{and}A\right)=\frac{\text{possible}\text{events}\text{for}\text{mushrrom}\text{and}\text{thick}\text{crust}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 2 for possible events for mushroom and thick crust pizza and 7 for total possible events in above formula.

$P\left(B\text{and}A\right)=\frac{2}{7}$

The probability for getting thick crust pizza is,

$P\left(A\right)=\frac{\text{possible}\text{event}\text{for}\text{thick}\text{crust}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 3 for possible event for thick crust pizza and 7 for total possible events in above formula.

$P\left(A\right)=\frac{3}{7}$

Substitute $\frac{2}{7}$ for $P\left(B\text{and}A\right)$ and $\frac{3}{7}$ for $P\left(A\right)$ in equation (2).

$\begin{array}{c}P\left(B|A\right)=\frac{\frac{2}{7}}{\frac{3}{7}}\\ =\frac{2}{3}\end{array}$

The probability for getting mushroom pizza is,

$P\left(B\right)=\frac{\text{possible}\text{event}\text{for}\text{mushroom}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 4 for possible event for mushroom pizza and 7 for total possible events in above formula.

$P\left(B\right)=\frac{4}{7}$

Substitute $\frac{2}{7}$ for $P\left(B\text{and}A\right)$ and $\frac{4}{7}$ for $P\left(B\right)$ in equation (1).

$\begin{array}{c}P\left(B|A\right)=P\left(B\right)\\ \frac{2}{3}\ne \frac{4}{7}\end{array}$

Thus, the events “getting a thick crust pizza” and “getting a pizza with mushrooms’ are not independent.

b.

To determine

Whether the events “getting a thick crust pizza” and “getting a pizza with mushrooms’ are independent or not.

Answer to Problem 13.37E

The events “getting a thick crust pizza” and “getting a pizza with mushrooms’ are independent.

Explanation of Solution

Given info:

Thomas’s pizza shop has seven pizzas in the oven, three out of seven are thick crust, and of these only one pizza has sausage and other two have mushrooms; remaining four pizzas have regular crust, and of these two pizza has only sausage and other two has only mushrooms. An eighth pizza is added to oven has thick crust with only cheese.

Calculation:

Let, A be the event for thick crust pizza and B be the event for mushroom pizza. Total number of pizzas is 8.

The formula for events to be independent,

$P\left(B|A\right)=P\left(B\right)$ (1)

The formula for conditional probability is,

$P\left(B|A\right)=\frac{P\left(B\text{and}A\right)}{P\left(A\right)}$ (2)

The probability for getting ‘mushroom and thick crust pizza’ is,

$P\left(B\text{and}A\right)=\frac{\text{possible}\text{events}\text{for}\text{mushrrom}\text{and}\text{thick}\text{crust}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 2 for possible events for mushroom and thick crust pizza and 8 for total possible events in above formula.

$P\left(B\text{and}A\right)=\frac{2}{8}$

The probability for getting thick crust pizza is,

$P\left(A\right)=\frac{\text{possible}\text{event}\text{for}\text{thick}\text{crust}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 4 for possible event for thick crust pizza and 8 for total possible events in above formula.

$\begin{array}{c}P\left(A\right)=\frac{4}{8}\\ =\frac{1}{2}\end{array}$

Substitute $\frac{2}{8}$ for $P\left(B\text{and}A\right)$ and $\frac{1}{2}$ for $P\left(A\right)$ in equation (2).

$\begin{array}{c}P\left(B|A\right)=\frac{\frac{2}{8}}{\frac{1}{2}}\\ =\frac{1}{2}\end{array}$

The probability for getting mushroom pizza is,

$P\left(B\right)=\frac{\text{possible}\text{event}\text{for}\text{mushroom}\text{pizza}}{\text{Total}\text{possible}\text{events}}$

Substitute 4 for possible event for mushroom pizza and 8 for total possible events in above formula.

$\begin{array}{c}P\left(B\right)=\frac{4}{8}\\ =\frac{1}{2}\end{array}$

Substitute $\frac{1}{2}$ for $P\left(B\text{and}A\right)$ and $\frac{1}{2}$ for $P\left(B\right)$ in equation (1).

$\begin{array}{c}P\left(B|A\right)=P\left(B\right)\\ \frac{1}{2}=\frac{1}{2}\end{array}$

Thus, the events “getting a thick crust pizza” and “getting a pizza with mushrooms” are independent.