Question

Mike’s pizza - You work at Mike’s pizza shop. You have the following information about the 9 pizzas in the oven: 3 of the 9 have thick crust and 2 of the 3 thick-crust pizzas have mushrooms. Of the remaining 6 pizzas, 4 have mushrooms.

a. Are the events “thick-crust pizza” and “pizza with mushrooms” mutually exclusive? Page Number: 356 Justify your answer.

b. Are the events “thick-crust pizza” and “pizza with mushrooms” independent? Justify your answer.

c. Suppose you randomly select 2 of the pizzas in the oven. Find the probability that both have mushrooms.

Short answer

(a) Events are not mutually exclusive .

(b) Events are not independent .

(c) Required probability is$\frac{5}{12}$.

Step-by-step solution

Part (a) - Step 1 : Given Information 

We are given that we have $9$pizzas in the oven: $3$of the $9$have thick crust and $2$of the $3$thick-crust pizzas have mushrooms. Of the remaining $6$pizzas , $4$have mushrooms . We need to find if the events “thick-crust pizza” and “pizza with mushrooms” mutually exclusive .

Part (a) - Step 2 : Explanation

For events to be mutually exclusive when $(A\cap B)=0$

The given events are not mutually exclusive because thick crust pizza have common things with mushroom . So the given events are not mutually exclusive .

Part (b) - Step 1 : Given Information 

We are given that we have $9$pizzas in the oven: $3$of the $9$have thick crust and $2$of the $3$thick-crust pizzas have mushrooms. Of the remaining $6$pizzas , $4$ have mushrooms . We are required to find if the events “thick-crust pizza” and “pizza with mushrooms” independent .

Part (b) - Step 2 : Explanation 

Let A be thick crust and Let B= mushrooms

P(A/B) =P(A)

$\frac{2}{3}\ne \frac{4}{7}$

P(B/A) = P(B)

$\frac{1}{2}\ne \frac{3}{7}$

Since the probabilities are not equal, these events are not independent .

Part (c) - Step 1 : Given Information

We are given $9$pizzas and we randomly select $2$of the pizzas in the oven. We need to find the probability that both pizzas have mushroom .

Part (c) - Step 2 : Explanation

The required probability is P (first pizza to be mushroom one )$\times$P(second mushroom pizza)

Required probability $=\frac{6}{9}\times \frac{5}{8}$

$=$$\frac{5}{12}$