Suppose 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 ft 
in height, and let B be the event that the selected individ-
ual is a professional basketball player. Which do you 
think is larger, P(AuB) or P(BuA)? Why?

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? 

---

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.

Suppose 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 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, P(AuB) or P(BuA)? Why?

Explore the probability concepts involved in comparing the likelihood of being over 6 feet tall or a professional basketball player. This problem delves into conditional probability and the use of Bayes' Theorem, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation