Mike’s pizza You work at Mike’s pizza shop. You have the following information about the $7$ pizzas in the oven: $3$ of the $7$ have thick crust, and of these $1$ has only sausage and $2$ have only mushrooms. The remaining $4$ pizzas have regular crust, and of these $2$ have only sausage and $2$ have only mushrooms.

Choose a pizza at random from the oven.

(a) Are the events {getting a thick-crust pizza} and {getting a pizza with mushrooms} independent? Explain.

(b) You add an eighth pizza to the oven. This pizza has thick crust with only cheese. Now are the events {getting a thick-crust pizza} and {getting a pizza with mushrooms} independent? Explain.

• Find the probability that an event occurs using a two-way table.

The total number of pizzas is seven.

$3$pizzas with a thick crust. $4$pizzas (normal size). In a thick crust pizza, the number of sausages equals $1$. Number of mushroom pizzas in thick crust pizza=$2$In a standard pizza, there are $2$sausages. Number of mushroom pizzas in a standard pizza=2.

$Theprobabilityofanevent=\frac{numberoffavourableoutcomes}{totalnumberofoutcomes}$

If two events A and B are independent,

then $P(A\cap B)=P(A\cap B|B)$

The information provided is summarized in the table below.

In that case, P (thick crust pizza with mushroom)=$2/7$

$P(thickcrustwithmushroom||thickcrustpizza)=\frac{Numberofthickcrustpizzawithmushroom}{Numberofthickcrustpizza}$

According to the table above, the number of thick crust mushroom pizzas is=$2$

Therefore, P (thick crust with mushroom thick crust pizza)=$\frac{2}{3}$

Because $\frac{2}{7}$ does not equal$\frac{2}{3}$, both events are not independent.

The table can be rearranged in the following manner:

So, total number of pizzas=$8$

Number of thick crust pizza=$4$

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

$P(thickcrustwithmushroom||thickcrustpizza)=\frac{numberofthickcrustpizzawithmushroom}{numberofthickcrustpizza}$

According to the table above,

$2$ thick-crust pizzas with mushroom topping

Therefore, $P(thickcrustwithmushroom||thickcrustpizza)=\frac{2}{4}$

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