Visa Card USA studied how frequently young consumers, ages 18 to 24, use plastic

Question

Visa Card USA studied how frequently young consumers, ages 18 to 24, use plastics (debit and credit) cards in making purchases. The results of the study provided the following probabilities.
* 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.
U.S. Census Bureau data show that 14% of the consumer population is 18 to 24 years old.
Given the consumer is over 24 years old, what is the probability that the consumer uses a plastic card?

Explanation / Answer

Given P(uses plastic) = 0.37, P(18 = age =24 | uses plastic) = 0.19 P(age > 24 | uses plastic) = 0.81 P(18 = age = 24) = 0.14 P(uses plastic | age > 24) = P(uses plastic & age > 24) / P(age > 24) = P(uses plastic) P(age > 24 | uses plastic) / P(age > 24) = (0.37) (0.81) / P(age > 24) = 0.2997 / P(age > 24). However, P(age > 24) isn't known. Since P(uses plastic & age > 24) = 0.2997, then this number serves as the greatest lower bound of P(age > 24), so P(age > 24) = 0.2997 (call this L). Since P(18 = age = 24) = 0.14, then the least upper bound of P(age > 24) would be if P(age < 18) = 0, so P(age > 24) = 0.86 (call this U). Therefore, since 0.2997 = P(age > 24) = 0.86 0.2997 / U = P(uses plastic | age > 24) = 0.2997 / L 0.2997 / 0.86 = P(uses plastic | age > 24) = 0.2997 / 0.2997 0.348488 = P(uses plastic | age > 24) = 1. In other words, without additional information, the best you can say is that the probability is at least 0.348 that a consumer uses a plastic card, given the consumer is over 24 years old.

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