Question

In a recent month, 88% of automobile drivers filled their vehicles with regular gasoline, 2% purchased midgrade gas, and 10% bought premium gas. Of those who bought regular gas, 28% paid with a credit card; of customers who bought midgrade and premium gas, 34% and 42%, respectively, paid with a credit card. Suppose we select a customer at random. Given that the customer paid with a credit card, find the probability that she bought premium gas. Show your work.

Answer 1

Answer: The required probability is 0.1422.

Explanation:

Since we have given that

Probability that drivers filled their vehicles with regular gasoline P(R) = 88%

Probability that drivers purchased midgrade gas P(M) = 2%

Probability that bought premium gas P(P) = 10%

Let A be the given event that it is paid with credit card.

Probability that who bought regular gas paid with credit card P(A|R) = 285

Probability that who bought midgrade gas with credit card P(A|M) = 34%

Probability that who bought premium gas with credit card P(A|P) = 42%

According to Bayes theorem, we get that

P(P|A) is given by

`(P(P).P(A|P))/(P(R).P(A|R)+P(M).P(A|M)+P(P).P(A|P))\\\\=(0.1* 0.42)/(0.88* 0.28+0.02* 0.34+0.1* 0.42)\\\\=(0.042)/(0.2952)\\\\=0.1422`

Hence, the required probability is 0.1422.