At a certain gas station, 50% of the customers use regular gas (A), 40% use plus

Question

At a certain gas station, 50% of the customers use regular gas (A), 40% use plus gas (A), and the remaining 10% use premium (A3). Of those customers getting regular gas only 70% completely fill (F) their tanks. Of the customers buying plus gas, 60% fill their tanks, and of the customers buying premium gas, 40% fill their tanks. What is the probability the next customer will fill their tank? a. b. If the next customer fills their tank, what is the probability they used regular gas? or P(4 F) c. Find P(A|F) d. Find P(4,|F)

Explanation / Answer

Here we are given that P(A1) = 0.5, P(A2) = 0.4 and P(A3) = 0.1

Also we are given that P(F | A1) = 0.7, P(F | A2) = 0.6 and P(F | A3) = 0.4

Note that F represents filled tank here.

a) By addition law of probability, probability that the next customer will fill their tank is computed as:

P(F) = P(F | A1)P(A1) + P(F | A2)P(A2) + P(F | A3)P(A3)

P(F) = 0.7*0.5 + 0.6*0.4 + 0.4*0.1 = 0.63

Therefore 0.63 is the required probability here.

b) Given that the customer filled the tank, probability that they used regular tank is computed using Bayes theorem as:

P( A1 | F) = P(F | A1)P(A1) / P(F) = 0.7*0.5/ 0.63 = 0.5556

Therefore 0.5556 is the required probability here.

c) Similarly other 2 probabilities are computed as:

P( A2 | F) = P(F | A2)P(A2) / P(F) =0.6*0.4/ 0.63 = 0.3810

Therefore 0.3810 is the required probability here.

d) Here we have:

P( A3 | F) = P(F | A3)P(A3) / P(F) =0.1*0.4/ 0.63 = 0.0635

Therefore 0.0635 is the required probability here.

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