To determine whether events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we need to understand the concept of independence in probability. Two events are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Given the problem:
- The probability that Roger wins the tournament, [tex]\( P(A) \)[/tex], is 0.45.
- The probability that Stephan wins the tournament, [tex]\( P(B) \)[/tex], is 0.40.
- The probability that Roger wins the tournament given that Stephan wins, [tex]\( P(A \mid B) \)[/tex], is 0.
- The probability that Stephan wins the tournament given that Roger wins, [tex]\( P(B \mid A) \)[/tex], is also given as 0.
We need to use this information to determine the dependence or independence of the events.
1. Check the given probabilities:
- [tex]\( P(A) = 0.45 \)[/tex]
- [tex]\( P(A \mid B) = 0 \)[/tex]
2. Determine if [tex]\( P(A \mid B) = P(A) \)[/tex]:
- We know that [tex]\( P(A \mid B) = 0 \)[/tex].
- We know that [tex]\( P(A) = 0.45 \)[/tex].
3. Compare [tex]\( P(A \mid B) \)[/tex] with [tex]\( P(A) \)[/tex]:
- Because [tex]\( P(A \mid B) = 0 \)[/tex] is not equal to [tex]\( P(A) = 0.45 \)[/tex], we conclude that [tex]\( P(A \mid B) \neq P(A) \)[/tex].
Since [tex]\( P(A \mid B) \neq P(A) \)[/tex], events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent. This leads directly to the conclusion:
The correct statement is:
C. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].
