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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?

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.

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