https://file.aiquickdraw.com/mathbot/file/1728670373435xkupe02y.jpg

Let's analyze and solve the problem from the image.

We are given:
- \( P(A) = 0.2 \)
- \( P(B) = 0.3 \)
- \( P(A \cup B) = 0.4 \)

We need to find the following:

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

We can use the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Substitute the known values:
\[
0.4 = 0.2 + 0.3 - P(A \cap B)
\]
\[
0.4 = 0.5 - P(A \cap B)
\]
\[
P(A \cap B) = 0.5 - 0.4 = 0.1
\]

### Part (b) \( P(\overline{A} \cup \overline{B}) \)

We use De Morgan's Law, which states:
\[
P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B)
\]
We already know that \( P(A \cap B) = 0.1 \), so:
\[
P(\overline{A} \cup \overline{B}) = 1 - 0.1 = 0.9
\]

### Part (c) \( P(\overline{A} \cap B) \)

The probability of \( \overline{A} \cap B \) can be found as:
\[
P(\overline{A} \cap B) = P(B) - P(A \cap B)
\]
Substitute the known values:
\[
P(\overline{A} \cap B) = 0.3 - 0.1 = 0.2
\]

### Part (d) \( P(\overline{A} | B) \)

We use the conditional probability formula:
\[
P(\overline{A} | B) = \frac{P(\overline{A} \cap B)}{P(B)}
\]
Substitute the known values:
\[
P(\overline{A} | B) = \frac{0.2}{0.3} = \frac{2}{3}
\]

### Summary of Results:
- (a) \( P(A \cap B) = 0.1 \)
- (b) \( P(\overline{A} \cup \overline{B}) = 0.9 \)
- (c) \( P(\overline{A} \cap B) = 0.2 \)
- (d) \( P(\overline{A} | B) = \frac{2}{3} \)

Would you like more details on any part, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How do you calculate \( P(\overline{A}) \) from \( P(A) \)?
2. Can you explain why we use De Morgan’s Law in part (b)?
3. What is the formula for conditional probability?
4. How can you verify the result for \( P(A \cap B) \) using a Venn diagram?
5. What happens to these probabilities if events A and B are independent?

**Tip**: When solving probability problems, always check if the events are mutually exclusive or independent, as this influences the formulas used!

Two events A and B are such that P(A) = 0.2, P(B) = 0.3, and P(A ∪ B) = 0.4. Find the following:
a) P(A ∩ B)
b) P(Ā ∪ B̅)
c) P(Ā ∩ B)
d) P(Ā | B)

Learn how to calculate the intersection and union of two events A and B, using probabilities P(A) = 0.2, P(B) = 0.3, and P(A ∪ B) = 0.4. This problem covers key probability concepts like De Morgan's laws and conditional probability, suitable for advanced high school or college-level students.

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