The expression \( P(A)P(B) - P(A \cap B) \) involves the probabilities of two events \( A \) and \( B \). To maximize this expression, we need to explore the behavior of probabilities.

Let’s break it down:

- \( P(A) \) and \( P(B) \) are the probabilities of events \( A \) and \( B \), respectively. Both \( P(A) \) and \( P(B) \) are between 0 and 1, i.e., \( 0 \leq P(A), P(B) \leq 1 \).
- \( P(A \cap B) \) is the probability of the intersection of \( A \) and \( B \), which satisfies \( 0 \leq P(A \cap B) \leq \min(P(A), P(B)) \).

To find the maximum of \( P(A)P(B) - P(A \cap B) \), consider the following special cases:

1. **Independence of Events**: When \( A \) and \( B \) are independent, \( P(A \cap B) = P(A)P(B) \). In this case:
   \[
   P(A)P(B) - P(A \cap B) = P(A)P(B) - P(A)P(B) = 0.
   \]
   So, the expression equals zero when \( A \) and \( B \) are independent.

2. **Disjoint Events**: When \( A \) and \( B \) are disjoint (mutually exclusive), \( P(A \cap B) = 0 \). In this case:
   \[
   P(A)P(B) - P(A \cap B) = P(A)P(B).
   \]
   This is the largest value since \( P(A \cap B) = 0 \), and the product \( P(A)P(B) \) can be maximized when \( P(A) = 1 \) and \( P(B) = 1 \), giving a maximum value of 1.

Thus, the maximum value of \( P(A)P(B) - P(A \cap B) \) is **1**, which occurs when \( A \) and \( B \) are disjoint and each has a probability of 1.

### Would you like more details on this or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How does \( P(A \cup B) \) change in terms of \( P(A) \) and \( P(B) \)?
2. What are the implications of \( P(A \cap B) = 0 \) in real-world events?
3. How do dependent events affect the maximum of this expression?
4. How does the expression behave if \( P(A) = 0 \) or \( P(B) = 0 \)?
5. Can you explain the relationship between joint probability and independent events?

**Tip**: For independent events, joint probabilities simplify as the product of their individual probabilities, which can help in many probability calculations!

Explore how to maximize the expression P(A)P(B) − P(A∩B) in probability theory. This detailed guide covers key concepts such as independent events, disjoint events, and the conditions that result in the maximum value of 1. Suitable for Grades 11-12 students learning advanced probability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation