Given the consumer is over 24 years old, what is the probability that the consumer uses a plastic card?
27 Feb 2010
* 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.
Dear adrjaboor,
The problem as stated doesn’t give enough information to find an exact answer.
From the problem statement,
P(uses plastic) = 0.37,
P(18 ≤ age ≤24 | uses plastic) = 0.19,
P(age > 24 | uses plastic) = 0.81, and
P(18 ≤ age ≤ 24) = 0.14,
where the vertical bar "|" stands for "given."
From the definition of conditional probability,
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, and can only be determined within a wide range. The reason is that it’s also possible for consumers to be under 18, and the problem says nothing directly about this category. 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.