You have the following information about 7 pizzas in the oven:3 of 7 pizza are thick crust, and of these one has only sausage and2 mushroom. the remaining four are regular crust, and of these 2have only sausage and 2 have only mushroom. 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.

Please help me :)

To answer the questions, we need to understand what it means for two events to be independent. Two events are considered independent if the occurrence of one does not affect the probability of the other happening. Let's explore each question individually:

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

To determine if these events are independent, we need to compare the probability of getting a thick crust pizza with the probability of getting a pizza with mushrooms.

From the information given -
- There are 3 thick crust pizzas.
- Of these 3, one has only sausage and two have mushrooms.
- There are 4 regular crust pizzas.
- Of these 4, 2 have only sausage and 2 have only mushrooms.

So, the probability of getting a thick crust pizza is 3/7, and the probability of getting a pizza with mushrooms is 2/7.

If the events were independent, the probability of getting a thick crust pizza with mushrooms would simply be the product of the individual probabilities. However, in this case, there are only two pizzas with mushrooms out of seven total pizzas. Therefore, the occurrence of the event "getting a thick crust pizza" affects the likelihood of "getting a pizza with mushrooms". Hence, the events are not independent.

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

With the addition of the eighth pizza, the information changes. Now we have:
- There are 4 thick crust pizzas in total (3 from before and the newly added one).
- Of these 4, two have mushrooms (as stated before).
- There are still 4 regular crust pizzas, with 2 having only sausage and 2 having only mushrooms.

So, the probability of getting a thick crust pizza remains 4/8, but the probability of getting a pizza with mushrooms remains 2/7.

In this case, again, the probability of getting a thick crust pizza with mushrooms is not equal to the product of the individual probabilities. The occurrence of the event "getting a thick crust pizza" still affects the likelihood of "getting a pizza with mushrooms". Hence, even with the additional pizza, these events are still not independent.

Therefore, both before and after adding the eighth pizza, the events (getting a thick crust pizza) and (getting a pizza with mushrooms) are not independent.

a. No, the events (getting a thick crust pizza) and (getting a pizza with mushrooms) are not independent. We can see that out of the 3 thick crust pizzas, 2 have mushrooms, which means that the probability of getting a thick crust pizza and a pizza with mushrooms are related. The probability of getting a thick crust pizza increases if we know that we are looking for a pizza with mushrooms.

b. After adding the eighth pizza, the events (getting a thick crust pizza) and (getting a pizza with mushrooms) are still not independent. Although the number of thick crust pizzas has increased to 4, we can see that 2 of them have mushrooms. This means that the probability of getting a thick crust pizza and a pizza with mushrooms are still related. The probability of getting a thick crust pizza with mushrooms has increased compared to before adding the eighth pizza.