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.
a. What is the probability that the next customer will request extra unleaded gas and fill the tank? State the rule used.
b. What is the probability that the next customer fills the tank?
c. If the next customer fills the tank, what is the probability that regular gas is not requested?

a) P(B ∩ F) = 0.21

b) P(F) = 0.455

c) P(A'|F) = 0.736

Step-by-step explanation:

We can consider the events

A = Regular Unleaded Gas

B = Extra Unleaded Gas

C = Premium Leaded Gas

F = Customer fills the tank

So,

P(A) = 0.4, P(B) = 0.35, P(C) = 0.25

P(F|A) = 0.3

P(F|B) = 0.6

P(F|C) = 0.5

a) This part is asking us to find the probability P(B ∩ F). For finding this out, we will use the conditional probability formula.

P(F|B) = P(B ∩ F)/P(B)

P(B ∩ F) = P(F|B) x P(B)

we have defined all the values above so we will just plug them in:

P(B ∩ F) = 0.6 x 0.35

P(B ∩ F) = 0.21

b) The probability that the next customer fills the tank can be computed by adding up the probabilities that the customer will request each type of fuel and fill the tank.

P(F) = P(A∩F) + P(B∩F) + P(C∩F)

     = P(F|A) x P(A) + P(F|B) x P(B) + P(F|C) x P(C)

     = 0.3x0.4 + 0.6x0.35 + 0.5x0.25

     = 0.12 + 0.21 + 0.125

P(F) = 0.455

c) If regular gas is not requested that means either extra gas or premium gas is requested. Given that the next customer fills the tank, the probability that regular gas is not requested is:

P(A'|F) = P(A'∩F)/P(F)

          = [P(B∩F) + P(C∩F)]/P(F)

          = [0.21 + 0.125]/0.455 (plugged in all values from part (b))

P(A'|F) = 0.736