Question

At a certain gas station, 40% of the customers use regular unleaded gas, 35% use extra unleaded gas, and 25% use premium unleaded gas. Of those customers using regular gas, only 30% fill their tanks. Of those customers using extra gas, 60% fill their tanks, whereas of those using premium, 50% fill their tanks.

- What is the probability that the next customer will request extra unleaded gas and fill the tank?
- What is the probability that the next customer fills the tank?
- If the next customer fills the tank, what is the probability that regular gas is requested? Extra gas? Premium gas?

Step 1

**Introduction:**

Define the following events:

A: The customer uses regular unleaded gas;

B: The customer uses extra unleaded gas;

C: The customer uses premium unleaded gas;

F: The customer fills their tank.

From the given percentages, the following probabilities are known:

P (A) = 0.40; P (B) = 0.35; P (C) = 0.25;

P (F | A) = 0.30; P (F | B) = 0.60; P (F | C) = 0.50.

Step 2

**Calculation:**

*Conditional probability:*

For two events M and N, the probability of the occurrence of M, given that event N has already occurred, is: P (M | N) = P (M ∩ N) / P (N).

From this conditional probability, it is evident that: P (M ∩ N) = P (M | N) P (N).

**Part 1:**

The probability that the next customer will request extra unleaded gas and fill the tank is P (B ∩ F). It is also known that P (B ∩ F) = P (F ∩ B). Using the above property of conditional probability, the calculation is done as follows:

P (B ∩ F)

= P (F ∩ B)

= P (F | B) P (B)

= (0.60) (0.35)

= **0.21**.

Thus, the probability that the next customer will request extra unleaded gas and fill the tank is ** 0.21**.

Step 3

**Part 2:**

The probability that the next customer fills the tank is P (F).

Now, P (F) = P (F ∩ A) + P (F ∩ B) + P (F ∩ C).

From the calculation of Part 1, P (F ∩ B) = 0.21. Using the same method as in Part 1, the calculations for P (F ∩ A) and P (F ∩ C) are done as follows:

P (F ∩ A)

= P (F | A) P (A)

= (0.30) (0.40)

= **0.12**.

P (F ∩ C)

= P (F...

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