Chapter 5, Problem 8CRE

(a)

To determine

To check: Whether the events of getting a thick crust pizza and pizza with mushrooms independent or not.

Answer to Problem 8CRE

No the events are not independent.

Explanation of Solution

Given:

Total number of pizzas $=7$

Number of thick crust pizza $=3$

Number of regular pizza $=4$

Number of sausage in thick crust pizza $=1$

Number of pizza with mushroom in thick crust pizza $=2$

Number of sausage in regular pizza $=2$

Number of pizza with mushroom in regular pizza $=2$

Formula used:

The probability of an event $=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

Condition for independence

If two events A and B are independent, then

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

Calculation:

The data given is described with the help of table below

	Sausage	Mushrooms	Total
Thick crust	1	2	3
Regular	2	2	4
Total	3	4	7

So,

P (thick crust pizza with mushroom) $=\frac{2}{7}$

  $\text{P }\left(\text{thick crust withmushroom }|\text{ thick crust pizza}\right)=\frac{\text{Number of thick crust pizza with mushroom}}{\text{Number of thick crust pizza}}$

Reading from the above table,

Number of thick crust pizza with mushroom $=2$

Therefore, $\text{P }\left(\text{thick crust withmushroom }|\text{ thick crust pizza}\right)=\frac{2}{3}$

As $\frac{2}{7}\ne \frac{2}{3}$ , so both the events are not independent.

(b)

To determine

To check: Whether the events of getting a thick crust pizza and pizza with mushrooms independent or not.

Answer to Problem 8CRE

The events of getting a thick crust pizza and pizza with mushrooms are independent.

Explanation of Solution

Calculation:

The table can be rearranged as follows

	Sausage	Mushrooms	Cheese	Total
Thick crust	1	2	1	4
Regular	2	2	0	4
Total	3	4	1	8

So, total number of pizzas $=8$

Number of thick crust pizza $=4$

P (thick crust pizza with mushroom) $=\frac{4}{8}$

  $\text{P }\left(\text{thick crust withmushroom }|\text{ thick crust pizza}\right)=\frac{\text{number of thick crust pizza with mushroom}}{\text{number of thick crust pizza}}$

Reading from the above table,

Number of thick crust pizza with mushroom $=2$

Therefore, $\text{P }\left(\text{thick crust withmushroom }|\text{ thick crust pizza}\right)=\frac{2}{4}$

As $\frac{4}{8}=\frac{2}{4}$, both events are independent.