* 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.
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.
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