Suppose that

*A*

and

*B*

are independent events such that

=*PA*0.40

and

=*PB*0.10

.

Find

*P*∩*AB*

and

*P*∪*AB*

. (If necessary, consult a list of formulas.)

(a)

=*P*∩*AB*

(b)

=*P*∪*AB*

To solve this problem, we need to calculate two probabilities related to independent events \( A \) and \( B \).

### Given:
- \( P(A) = 0.40 \)
- \( P(B) = 0.10 \)
- \( A \) and \( B \) are independent events.

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

Since \( A \) and \( B \) are independent events, the probability of both \( A \) and \( B \) happening (i.e., \( P(A \cap B) \)) is the product of their individual probabilities:

\[
P(A \cap B) = P(A) \times P(B)
\]

Substitute the given values:

\[
P(A \cap B) = 0.40 \times 0.10 = 0.04
\]

### (b) \( P(A \cup B) \) 

The probability of either \( A \) or \( B \) or both occurring (i.e., \( P(A \cup B) \)) can be calculated using the formula:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Substitute the values:

\[
P(A \cup B) = 0.40 + 0.10 - 0.04 = 0.46
\]

### Final Answers:

- **(a)** \( P(A \cap B) = 0.04 \)
- **(b)** \( P(A \cup B) = 0.46 \)

Would you like more details on this? 

Here are 5 related questions you might find helpful:
1. What is the general formula for the union of two events?
2. How does the concept of independence affect the calculation of joint probabilities?
3. What if \( A \) and \( B \) were not independent—how would that change the calculation?
4. Can the probability of \( P(A \cup B) \) ever be greater than 1?
5. What is the difference between mutually exclusive and independent events?

**Tip:** Understanding the difference between independent and mutually exclusive events is crucial for correctly applying probability formulas.

Learn how to calculate the probability of intersection and union for independent events A and B with a step-by-step solution. Understand the formulas involved and their application in probability theory.

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