Question

Suppose that an individual is randomly selected from the population of all adult males living in the United States. Let \(A\) be the event that the selected individual is over 6 feet in height, and let \(B\) be the event that the selected individual is a professional basketball player. Which do you think is larger, \(P(A \mid B)\) or \(P(B \mid A) ?\) Why?

Short answer

The probability that a randomly chosen professional basketball player is over 6 feet tall (\(P(A \mid B)\)) is larger than the probability that a randomly chosen individual who is over 6 feet tall is a professional basketball player (\(P(B \mid A)\)). This is because the proportion of professional basketball players who are over 6 feet tall is very high, while the proportion of people over 6 feet tall who are professional basketball players is relatively small.

Step-by-step solution

Understanding the Probabilities

First, let's think about what these probabilities mean:\n\n1. \(P(A \mid B)\) is the probability that a randomly chosen professional basketball player is over 6 feet tall. Since professional basketball tends to favor taller individuals, this probability is likely quite high.\n\n2. \(P(B \mid A)\) is the probability that a randomly chosen individual who is over 6 feet tall is a professional basketball player. Although being tall could certainly be an advantage in basketball, there are many individuals who are over 6 feet tall and are not professional basketball players. Therefore, this probability is probably quite low.

Comparing the Probabilities

After understanding the two probabilities, comparing them will lead to a relatively clear result. \(P(A \mid B)\) is very likely to be larger than \(P(B \mid A)\). This is because that within the population of professional basketball players, the proportion that is over 6 feet is large, while within the population of males over 6 feet, the proportion that are professional basketball players is relatively small.

Understanding the Reason

The main reason for this difference in the probabilities boils down to the size of different populations. The total population of people over 6 feet is much larger than the total population of professional basketball players. Hence, even though being over 6 feet tall is a common characteristic among professional basketball players, professional basketball players only make up a small fraction of all people who are over 6 feet tall.