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Let's solve the questions step by step based on the information provided:

We are given:
- \( P(A) = 0.5 \)
- \( P(B) = 0.52 \)
- A and B are independent events.

### a) Find \( P(A \cap B) \) (the probability of both A and B happening):
Since A and B are independent, we use the formula:
\[
P(A \cap B) = P(A) \times P(B)
\]
\[
P(A \cap B) = 0.5 \times 0.52 = 0.26
\]
So, \( P(A \cap B) = 0.26 \).

### b) Find \( P(A \cup B) \) (the probability of A or B happening):
We use the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
We already calculated \( P(A \cap B) = 0.26 \), so:
\[
P(A \cup B) = 0.5 + 0.52 - 0.26 = 0.76
\]
Thus, \( P(A \cup B) = 0.76 \).

### c) Find \( P(A | B) \) (the conditional probability of A given B):
We use the formula for conditional probability:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Substituting the known values:
\[
P(A | B) = \frac{0.26}{0.52} = 0.5
\]
So, \( P(A | B) = 0.5 \).

### d) Find \( P(B | A) \) (the conditional probability of B given A):
Again, we use the conditional probability formula:
\[
P(B | A) = \frac{P(A \cap B)}{P(A)}
\]
Substituting the values:
\[
P(B | A) = \frac{0.26}{0.5} = 0.52
\]
Thus, \( P(B | A) = 0.52 \).

### Summary of answers:
a) \( P(A \cap B) = 0.26 \)

b) \( P(A \cup B) = 0.76 \)

c) \( P(A | B) = 0.5 \)

d) \( P(B | A) = 0.52 \)

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Would you like further details on any of these calculations?

Here are 5 related questions to consider:
1. How do we know when events are independent?
2. How would the calculations change if events were dependent?
3. What is the relationship between conditional probability and independence?
4. Can two events be both mutually exclusive and independent?
5. How would we calculate \( P(A^c \cap B^c) \), where \( A^c \) and \( B^c \) are the complements of A and B?

**Tip:** Independence means that the occurrence of one event does not affect the probability of the other event. This simplifies calculations like \( P(A \cap B) \).

Given the following information, answer questions a - d. P(A) = 0.5, P(B) = 0.52. A and B are independent. Round all answers to 5 decimal places as needed. a) Find P(A ∩ B). b) Find P(A ∪ B). c) Find P(A | B). d) Find P(B | A).

Learn how to calculate probabilities for independent events using given values of P(A) = 0.5 and P(B) = 0.52. This problem involves calculating P(A ∩ B), P(A ∪ B), P(A | B), and P(B | A) using standard probability formulas.

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