At a certain gas station 40 of the customers use regular unl

At a certain gas station, 40% of the customers use regular unleaded gas (A₁), 35% use extra unleaded gas (A₂), and 25% use premium unleaded gas (A₃). Of those customers using regular gas, only 30% fill their tanks (event B). 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?

(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 requested? Extra gas? Premium gas?

Solution

At a certain gas station, 40% of the customers use regular unleaded gas (A₁), 35% use extra unleaded gas (A₂), and 25% use premium unleaded gas (A₃). Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using extra gas, 60% fill their tanks, whereas of those using premium, 50% fill their tanks.

P( A1)=0.4   P(A2)=0.35   P(A3)=0.25

P(B/A1) =0.3   P(B/A2)=0.6   P(B/A3) =0.5

P(A2 AND B) = P(B/A2)* P(A2)=0.6*0.35=0.21

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

    =0.3*0.4+0.6*0.35+0.5*0.25 = 0.455

P(A1/B) =P(B/A1)* P( A1)/P(B)

=0.3*0.4/0.455 =0.352

Extra gas?

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

=0.6*0.35/0.455 =0.462

Premium gas?

   P(A3/B) =P(B/A3)* P( A3)/P(B)

=0.5*0.25/0.455 =0.275