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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.

Respuesta :

Answer: The required probability is 0.1422.

Step-by-step 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

[tex]\dfrac{P(P).P(A|P)}{P(R).P(A|R)+P(M).P(A|M)+P(P).P(A|P)}\\\\=\dfrac{0.1\times 0.42}{0.88\times 0.28+0.02\times 0.34+0.1\times 0.42}\\\\=\dfrac{0.042}{0.2952}\\\\=0.1422[/tex]

Hence, the required probability is 0.1422.

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