Suppose A and B are events such that P(A) = 0.34 and P(B) = 0.57 .
Assuming that events A and B are independent, help Kori answer the following questions. You may round your answers to 2 decimal places, and use previous values in subsequent computations.
Showing work is REQUIRED.
1. P(ANB) =
2. P(AUB) =
3. P(A |B) =
4. P(A' В) =

Let's go through these step by step, using the provided values and showing the necessary work for each part.

### Given:
- \( P(A) = 0.34 \)
- \( P(B) = 0.57 \)
- Events \( A \) and \( B \) are independent.

For independent events, we know the following rules:
- \( P(A \cap B) = P(A) \times P(B) \) for independent events.
- \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
- \( P(A' \cap B) = P(B) - P(A \cap B) \).
- \( P(A | B) = \frac{P(A \cap B)}{P(B)} \).

Now let's solve the problems one by one.

---

### 1. \( P(A \cap B) \) (the intersection of A and B):
Since \( A \) and \( B \) are independent events, we can compute the intersection as:
\[
P(A \cap B) = P(A) \times P(B) = 0.34 \times 0.57 = 0.1938
\]
So, 
\[
P(A \cap B) \approx 0.19
\]

---

### 2. \( P(A \cup B) \) (the union of A and B):
The formula for the union of two events is:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Substitute the known values:
\[
P(A \cup B) = 0.34 + 0.57 - 0.19 = 0.91 - 0.19 = 0.72
\]
So, 
\[
P(A \cup B) = 0.72
\]

---

### 3. \( P(A | B) \) (the conditional probability of A given B):
The conditional probability formula is:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Substitute the known values:
\[
P(A | B) = \frac{0.19}{0.57} \approx 0.3333
\]
So, 
\[
P(A | B) \approx 0.33
\]

---

### 4. \( P(A' \cap B) \) (the intersection of the complement of A and B):
The complement of \( A \), denoted as \( A' \), means the probability of not \( A \). The formula for this is:
\[
P(A' \cap B) = P(B) - P(A \cap B)
\]
Substitute the known values:
\[
P(A' \cap B) = 0.57 - 0.19 = 0.38
\]
So, 
\[
P(A' \cap B) = 0.38
\]

---

### Summary of answers:
1. \( P(A \cap B) = 0.19 \)
2. \( P(A \cup B) = 0.72 \)
3. \( P(A | B) = 0.33 \)
4. \( P(A' \cap B) = 0.38 \)

---

Would you like more details on any of the steps? Here are five related questions for further exploration:

1. How would the results change if \( A \) and \( B \) were **not** independent?
2. Can you compute \( P(B' \cap A) \)?
3. What is the probability of \( A' \cup B' \)?
4. How would you find the probability of \( P(A | A \cup B) \)?
5. What would \( P(A' | B) \) represent and how would you calculate it?

**Tip**: Remember that conditional probabilities can be interpreted as the probability of one event happening, given that another has occurred. It's a key concept in understanding dependent relationships between events!

Suppose A and B are events such that P(A) = 0.34 and P(B) = 0.57. Assuming that events A and B are independent, help Kori answer the following questions. You may round your answers to 2 decimal places, and use previous values in subsequent computations.
1. P(ANB) = 
2. P(AUB) = 
3. P(A |B) = 
4. P(A' В) = 

Learn how to compute the probabilities of independent events using key probability formulas. This example explores P(A ∩ B), P(A ∪ B), P(A | B), and P(A' ∩ B), with step-by-step solutions. Suitable for Grades 10-12.

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