Event A occurs with probability 0.2. Event B occurs with probability 0.78. Events A and B are independent.
Find: a) P(An B)
b) P(AUB)
c) P(A| B)
d) P(A'UB)

We are given the following probabilities:

- \( P(A) = 0.2 \)
- \( P(B) = 0.78 \)
- Events \( A \) and \( B \) are independent.

Now let's solve each part:

### a) \( P(A \cap B) \) (The probability of both events A and B occurring)
Since \( A \) and \( B \) are independent, the probability of both events occurring is the product of their individual probabilities:
\[
P(A \cap B) = P(A) \times P(B)
\]
\[
P(A \cap B) = 0.2 \times 0.78 = 0.156
\]

### b) \( P(A \cup B) \) (The probability of either A or B occurring)
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:
\[
P(A \cup B) = 0.2 + 0.78 - 0.156 = 0.824
\]

### c) \( P(A \mid B) \) (The conditional probability of A given B)
The conditional probability is calculated as:
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]
Substitute the known values:
\[
P(A \mid B) = \frac{0.156}{0.78} = 0.2
\]

### d) \( P(A' \cup B) \) (The probability of either A not occurring or B occurring)
Using the complement rule, \( P(A') = 1 - P(A) \):
\[
P(A') = 1 - 0.2 = 0.8
\]
Now calculate \( P(A' \cup B) \) using the formula for the union:
\[
P(A' \cup B) = P(A') + P(B) - P(A' \cap B)
\]
Since \( A' \) and \( B \) are independent, \( P(A' \cap B) = P(A') \times P(B) \):
\[
P(A' \cap B) = 0.8 \times 0.78 = 0.624
\]
Now substitute into the formula:
\[
P(A' \cup B) = 0.8 + 0.78 - 0.624 = 0.956
\]

### Final answers:
a) \( P(A \cap B) = 0.156 \)  
b) \( P(A \cup B) = 0.824 \)  
c) \( P(A \mid B) = 0.2 \)  
d) \( P(A' \cup B) = 0.956 \)

Would you like further details or clarification?

Here are 5 related questions for further practice:
1. What is the probability of neither A nor B occurring?
2. How would you find \( P(B \mid A) \)?
3. What is \( P(A' \cap B') \)?
4. If \( P(A) \) were different, how would that affect \( P(A \cup B) \)?
5. What would change if events A and B were dependent?

**Tip:** For independent events, the intersection probability is simply the product of individual probabilities, making the calculations straightforward!

Event A occurs with probability 0.2. Event B occurs with probability 0.78. Events A and B are independent. Find: a) P(An B) b) P(AUB) c) P(A| B) d) P(A'UB)

Explore the probability of independent events A and B, covering calculations for intersections, unions, conditional probability, and complementary events. This problem uses key formulas like P(A ∩ B), P(A ∪ B), and P(A' ∪ B) and is suitable for high school students in Grades 10-12.

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