At a certain gas station, 40% of the customers use regulargas(A1), 35 % use plus gas (A2) and 25 % usepremium (A3). Of those customers using regular gas, only30 % fill their tanks(event B). Of those customers using plus, 60 %fill their their tanks, whereas of those premium, 50 % fill theretanks.
a) What is the probability that the next customer will requestplus gas and fill their tank ?
b) What is the probability that the next customer fills thetank ?
c)If the next customer fills the tank, what is the probabilitythat the regular gas is requested? Plus ?Premium

a)  P(A2 & B) = 0.21  

b)  P(B) = 0.455

c) P(A1/B) = 0.2637 , P(A2/B) = 0.4615 ,   P(A3/B) = 0.2747

Step-by-step explanation:

Given:

- P (A1) = 0.40

- P (A2) = 0.35

- P (A3) = 0.25

- P(B / A1) = 0.3

- P(B / A2) = 0.6

- P(B / A3) = 0.5

Find:

a) What is the probability that the next customer will request plus gas and fill their tank ?

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 the regular gas is requested? Plus ?Premium

Solution:

a)

- The probability that the next customer fill plus gas for complete tank is:

                         P(A2 & B) = P(A2)*P(B/A2)

                         P(A2 & B) = 0.35*0.6

                        P(A2 & B) = 0.21  

b)

- The probability that the next customer fill complete tank is:

                         P(B) = P(A2)*P(B/A2) + P(A1)*P(B/A1) + P(A3)*P(B/A3)

                         P(B) = 0.21 + 0.4*0.3 + 0.25*0.5

                         P(B) = 0.455

c)

- The probability that the next customer fill complete tank what is the probability that each of the gas is requested

                         P(A1/B) = P(A1 & B)/ P(B)

                         P(A1/B) = 0.4*0.3/0.455

                         P(A1/B) = 0.2637

                         P(A2/B) = P(A2 & B)/ P(B)

                         P(A2/B) = 0.21/0.455

                         P(A2/B) = 0.4615

                         P(A3/B) = P(A3 & B)/ P(B)

                         P(A3/B) = 0.25*0.5/0.455

                         P(A3/B) = 0.2747

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