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The problem asks you to complete the table using the multiplication rule for probabilities, which is:

\[
P(A \cap B) = P(A) \cdot P(B|A)
\]
or equivalently,
\[
P(A \cap B) = P(B) \cdot P(A|B)
\]

Let me fill in the missing values based on the given data and the multiplication rule.

### First row:
- \( P(A) = 0.8 \), \( P(B) = 0.5 \), and \( P(A|B) = 0.4 \)
  - \( P(A \cap B) = P(B) \cdot P(A|B) = 0.5 \cdot 0.4 = 0.2 \)
  - \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.2}{0.8} = 0.25 \)

### Second row:
- \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(B|A) = 0.2 \)
  - \( P(A \cap B) = P(A) \cdot P(B|A) = 0.5 \cdot 0.2 = 0.1 \)
  - \( P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.4} = 0.25 \)

### Third row:
- \( P(A) = 0.8 \), \( P(A \cap B) = 0.5 \), and \( P(B|A) = 0.25 \)
  - \( P(B) = \frac{P(A \cap B)}{P(A|B)} = \frac{0.5}{0.25} = 2 \) → Check data (wrong value?)

### Fourth row:
- \( P(A) = 0.6 \), \( P(B) = 0.2 \), and \( P(A|B) = 0.5 \)
  - \( P(A \cap B) = P(B) \cdot P(A|B) = 0.2 \cdot 0.5 = 0.1 \)
  - \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.1}{0.6} \approx 0.17 \)

### Fifth row:
- \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.1 \)
  - \( P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.5} = 0.2 \)
  - \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.1}{0.4} = 0.25 \)

### Sixth row:
- \( P(A) = 0.4 \), \( P(B) = 0.4 \), and \( P(A|B) = 0.6 \)
  - \( P(A \cap B) = P(B) \cdot P(A|B) = 0.4 \cdot 0.6 = 0.24 \)
  - \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.24}{0.4} = 0.6 \)

Let me know if you want further clarifications or details.

### Related Questions:
1. What is the significance of \( P(A \cap B) \) in probability theory?
2. How is the conditional probability \( P(B|A) \) calculated?
3. Can \( P(A|B) \) ever be greater than \( P(A) \)?
4. What happens when \( P(A \cap B) = 0 \)? What does this imply?
5. How would you interpret the multiplication rule in terms of independent events?

### Tip:
Always check the consistency of the data in probability problems, as incorrect inputs can lead to contradictions in the results.

Lad A og B være to hændelser i et udfaldsrum U. Benyt multiplikationsloven til at udfylde følgende skema.

Solve this probability problem by applying the multiplication rule to find the values of P(A ∩ B), P(A|B), and P(B|A) based on given probabilities for events A and B. The table requires calculations using conditional probability formulas.

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