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To answer the probability questions using the contingency table, let me break down the data.

From the table:

- **A ∩ B** (both A and B happen) = 20
- **A ∩ B'** (A happens and B does not) = 30
- **A' ∩ B** (A does not happen and B does) = 30
- **A' ∩ B'** (neither A nor B happens) = 50

Now, let's calculate the probabilities:

### Total number of outcomes:
\[
\text{Total} = 20 + 30 + 30 + 50 = 130
\]

### a. \( P(A | B) \) (Probability of A given B)
The conditional probability formula is:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

- \( P(A \cap B) = \frac{20}{130} \)
- \( P(B) = \frac{20 + 30}{130} = \frac{50}{130} \)

\[
P(A | B) = \frac{\frac{20}{130}}{\frac{50}{130}} = \frac{20}{50} = 0.4
\]

### b. \( P(A | B') \) (Probability of A given B')
\[
P(A | B') = \frac{P(A \cap B')}{P(B')}
\]

- \( P(A \cap B') = \frac{30}{130} \)
- \( P(B') = \frac{30 + 50}{130} = \frac{80}{130} \)

\[
P(A | B') = \frac{\frac{30}{130}}{\frac{80}{130}} = \frac{30}{80} = 0.375
\]

### c. \( P(A' | B') \) (Probability of A' given B')
\[
P(A' | B') = \frac{P(A' \cap B')}{P(B')}
\]

- \( P(A' \cap B') = \frac{50}{130} \)

\[
P(A' | B') = \frac{\frac{50}{130}}{\frac{80}{130}} = \frac{50}{80} = 0.625
\]

### d. Are events A and B independent?

To check for independence, we verify if:
\[
P(A \cap B) = P(A) \times P(B)
\]

- \( P(A \cap B) = \frac{20}{130} = 0.1538 \)
- \( P(A) = \frac{20 + 30}{130} = \frac{50}{130} = 0.3846 \)
- \( P(B) = \frac{20 + 30}{130} = \frac{50}{130} = 0.3846 \)

Now, check if \( P(A \cap B) = P(A) \times P(B) \):
\[
P(A) \times P(B) = 0.3846 \times 0.3846 = 0.1478
\]
Since \( 0.1538 \neq 0.1478 \), **A and B are not independent**.

---

Would you like more details on any step, or do you have any questions?

Here are 5 related questions you could explore:
1. How do we calculate the joint probability of A and B?
2. What is the general condition for two events to be independent?
3. How would the table change if A and B were independent?
4. Can you explain the concept of conditional probability in more detail?
5. What happens if all probabilities in a contingency table are equal?

**Tip**: When dealing with conditional probabilities, always check if you need the total probability of the condition (denominator) to be sure you're calculating correctly.

Use the contingency table below to find the probability of the following events: a. A|B, b. A|B', c. A'|B', d. Are events A and B independent?

Learn how to calculate conditional probabilities such as P(A|B), P(A|B'), and P(A'|B') using a contingency table. Also, explore the conditions for determining whether events A and B are independent. This problem is suitable for students in Grades 10-12.

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