To determine whether events \( A \) and \( B \) are independent, we need to check if the probability of \( A \) given \( B \) is equal to the probability of \( A \), and if the probability of \( B \) given \( A \) is equal to the probability of \( B \).
Step-by-Step Solution:
1. Given Information:
- \( P(A) = 0.45 \): The probability that Roger wins.
- \( P(B) = 0.40 \): The probability that Stephan wins.
- \( P(A \mid B) = 0.00 \): The probability that Roger wins given that Stephan wins.
- \( P(B \mid A) = 0.00 \): The probability that Stephan wins given that Roger wins.
2. Independence Criteria:
Events \( A \) and \( B \) are independent if:
[tex]\[
P(A \mid B) = P(A) \quad \text{and} \quad P(B \mid A) = P(B)
\][/tex]
3. Check the Given Probabilities:
- \( P(A \mid B) = 0.00 \) and \( P(A) = 0.45 \):
Since \( P(A \mid B) \neq P(A) \), the first condition for independence is not met.
- \( P(B \mid A) = 0.00 \) and \( P(B) = 0.40 \):
Since \( P(B \mid A) \neq P(B) \), the second condition for independence is also not met.
4. Conclusion:
Since neither condition for independence is satisfied, events \( A \) and \( B \) are not independent.
Therefore, the correct statement is:
D. Events \( A \) and \( B \) are not independent because \( P(A \mid B) \neq P(A) \).
Hence, the answer is D.
