https://file.aiquickdraw.com/mathbot/file/1728762663880t5i285qr.jpg

To solve this problem, we need to find \( P(A \cap B) \) and \( P(A' \cup B) \), given the following information:

1. \( P(A) = 0.2 \)
2. \( P(B) = 0.4 \)
3. \( P(A \cup B) = 0.5 \)

Let's solve each part step-by-step:

### Part (a): Finding \( P(A \cap B) \)

We use the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Substitute the given values into the formula:
\[
0.5 = 0.2 + 0.4 - P(A \cap B)
\]

Simplify and solve for \( P(A \cap B) \):
\[
0.5 = 0.6 - P(A \cap B)
\]
\[
P(A \cap B) = 0.6 - 0.5
\]
\[
P(A \cap B) = 0.1
\]

### Part (b): Finding \( P(A' \cup B) \)

We use the formula for the complement of \( A \):
\[
P(A') = 1 - P(A)
\]
Substitute the given value for \( P(A) \):
\[
P(A') = 1 - 0.2 = 0.8
\]

Now, we need to find \( P(A' \cup B) \), which can be calculated using the formula for the union of two events:
\[
P(A' \cup B) = P(A') + P(B) - P(A' \cap B)
\]

To find \( P(A' \cap B) \), we use the fact that \( P(A' \cap B) = P(B) - P(A \cap B) \) (since \( A' \cap B \) represents the part of \( B \) that does not overlap with \( A \)):
\[
P(A' \cap B) = 0.4 - 0.1 = 0.3
\]

Now, substitute back into the formula for \( P(A' \cup B) \):
\[
P(A' \cup B) = 0.8 + 0.4 - 0.3
\]
\[
P(A' \cup B) = 1.2 - 0.3 = 0.9
\]

### Final Answers:
- **(a)** \( P(A \cap B) = 0.1 \)
- **(b)** \( P(A' \cup B) = 0.9 \)

Would you like any additional explanations or details on these calculations?

Here are some related questions to explore further:
1. How would you find \( P(A \cup B') \) given the same information?
2. What is the probability of \( P(A' \cap B') \)?
3. How would these probabilities change if \( A \) and \( B \) were independent events?
4. Can you use Venn diagrams to illustrate these probabilities?
5. How would you find \( P(A | B) \) and \( P(B | A) \) with this information?

**Tip:** Always verify the consistency of probability values to ensure that the solution is correct, especially when dealing with complements and intersections.

If P(A) = 0.2 and P(B) = 0.4 and P(A ∪ B) = 0.5, find: a) P(A ∩ B) b) P(A' ∪ B)

Learn how to solve a probability problem involving P(A) = 0.2, P(B) = 0.4, and P(A ∪ B) = 0.5. This solution explains how to find P(A ∩ B) and P(A' ∪ B) using set theory and probability formulas. Suitable for high school students (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