Visa Card USA studied how frequently young consumers, ages 18 to 24, use plastic (debit and credit) cards in making purchases (Associated Press, January 16, 2006). The results of the study provided the following probabilities. (Note: You may assume that Visa only issues credit cards to individuals 18 years or older.) U.S. Census Bureau data show that 14% of the consumer population is 18 to 24 years old. The probability that a consumer uses a plastic card when making a purchase is .37. Given that the consumer uses a plastic card, there is a .19 probability that the consumer is 18 to 24 years old. Given that the consumer uses a plastic card, there is a .81 probability that the consumer is more than 24 years old. Given the consumer is 18 to 24 years old, what is the probability that the consumer uses a plastic card (to 4 decimals)?

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

In this problem, we have:

What is the probability of the consumer using a plastic card, given that the consumer is 18 to 24 years old.

The problem states that the probability that a consumer uses a plastic card when making a purchase is .37, so [tex]P(B) = 0.37[/tex]

P(A/B) is the probability that the consumer being 18 to 24 years old, given that he uses a plastic card. The problem states that this probability is .19. So [tex]P(A/B) = 0.19[/tex]

P(A) is the probability that the consumer is 18 to 24 years old. There is a .81 probability that the consumer is more than 24 years old. So there is a .19 probability that he is 18 to 24 years old. So [tex]P(A) = 0.19[/tex]

The probability is

[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.37*0.19}{0.19} = 0.37[/tex]

Given the consumer is 18 to 24 years old, there is 37% probability that he uses a plastic card.