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.