To analyze the problem, let's define the events more clearly:

• $A$: The event that a randomly selected individual is over 6 feet in height.
• $B$: The event that a randomly selected individual is a professional basketball player.

You are comparing $P(A \cup B)$ and $P(B \cup A)$, but these are the same thing because $A \cup B = B \cup A$. This represents the probability that the selected individual is either over 6 feet in height or a professional basketball player (or both).

However, I believe the question is actually asking you to compare conditional probabilities:

• $P(A | B)$: The probability that the individual is over 6 feet tall, given that they are a professional basketball player.
• $P(B | A)$: The probability that the individual is a professional basketball player, given that they are over 6 feet tall.

Intuitive Answer:

• $P(A | B)$: The vast majority of professional basketball players are over 6 feet tall. Thus, $P(A | B)$ is very large, close to 1.

• $P(B | A)$: While many men over 6 feet tall exist, only a tiny fraction of them are professional basketball players. Thus, $P(B | A)$ is very small.

Conclusion:

• $P(A | B)$ (the probability that the individual is over 6 feet tall given that they are a professional basketball player) is much larger than $P(B | A)$ (the probability that the individual is a professional basketball player given that they are over 6 feet tall).

The reasoning is that being a professional basketball player almost guarantees that you're over 6 feet tall, but being over 6 feet tall does not guarantee you're a professional basketball player.

Would you like more details on how to calculate conditional probabilities?

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Here are 5 related questions to explore further:

1. How do you calculate the conditional probability $P(A | B)$ using Bayes' Theorem?
2. What is the difference between mutually exclusive and independent events?
3. How would $P(A \cap B)$ (the probability of being both over 6 feet and a professional basketball player) be determined?
4. How do we compute $P(A \cup B)$ if we know the probabilities of $A$, $B$, and $A \cap B$?
5. How does the total probability of $A \cup B$ change if $A$ and $B$ are independent events?

Tip: Conditional probability is a key concept in statistics, often used in fields such as medicine, sports, and decision-making to assess the likelihood of events given prior information.