**Given the following information, answer questions e - g.**
P(A)=0.47
P(B)=0.5
A and B are dependent.
P(A∣B)=0.468
*Round all answers to 5 decimal places as needed*

**e) Find P(A∩B).**  

**f) Find P(A∪B).**  

**g) Find P(B∣A)**

Let's begin by using the given information to calculate the answers for questions e - g.

### Given Information:
- \( P(A) = 0.47 \)
- \( P(B) = 0.5 \)
- \( A \) and \( B \) are dependent.
- \( P(A \mid B) = 0.468 \)

Since \( A \) and \( B \) are dependent, we can use the conditional probability formula to find the joint probability.

### **e) Find \( P(A \cap B) \)**

The conditional probability formula is:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

We need to solve for \( P(A \cap B) \):

\[
P(A \cap B) = P(A \mid B) \cdot P(B)
\]

Substituting the given values:

\[
P(A \cap B) = 0.468 \times 0.5 = 0.234
\]

**Answer: \( P(A \cap B) = 0.23400 \)** (rounded to 5 decimal places)

### **f) Find \( P(A \cup B) \)**

The formula for the union of two events is:

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

Substituting the known values:

\[
P(A \cup B) = 0.47 + 0.5 - 0.234
\]

\[
P(A \cup B) = 0.736
\]

**Answer: \( P(A \cup B) = 0.73600 \)** (rounded to 5 decimal places)

### **g) Find \( P(B \mid A) \)**

The conditional probability of \( B \) given \( A \) is given by:

\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\]

Substituting the known values:

\[
P(B \mid A) = \frac{0.234}{0.47}
\]

\[
P(B \mid A) = 0.49787
\]

**Answer: \( P(B \mid A) = 0.49787 \)** (rounded to 5 decimal places)

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### Do you have any questions or need further details?

### Related Questions:
1. How would you calculate \( P(A \cap B) \) if \( A \) and \( B \) were independent?
2. What is the difference between dependent and independent events?
3. How can the union formula be used for mutually exclusive events?
4. How would you determine \( P(A^c \cap B) \), the probability that \( A \) does not occur but \( B \) does?
5. How do conditional probabilities change if \( P(B) = 0 \)?

### Tip:
Always remember that when dealing with dependent events, conditional probabilities provide a way to understand how the occurrence of one event affects the likelihood of another.

Given the following information, answer questions e - g. P(A)=0.47, P(B)=0.5, A and B are dependent, P(A∣B)=0.468. Round all answers to 5 decimal places as needed. e) Find P(A∩B). f) Find P(A∪B). g) Find P(B∣A).

Learn how to solve a probability problem involving dependent events. This problem includes calculating P(A∩B), P(A∪B), and P(B|A) using given values of P(A), P(B), and P(A|B). Suitable for students in Grades 10-12, it covers concepts such as conditional probability, intersection, and union of events.

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