To determine the relationship between events [tex]\( B \)[/tex] and [tex]\( C \)[/tex], we need to analyze the given probabilities: [tex]\( P(B) = 0.75 \)[/tex], [tex]\( P(B \text{ AND } C) = 0 \)[/tex], [tex]\( P(C) = 0.55 \)[/tex], and [tex]\( P(B \mid C) = 0 \)[/tex].
1. Checking if the events are Mutually Exclusive:
Events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are mutually exclusive if they cannot occur simultaneously. This is mathematically represented as [tex]\( P(B \text{ AND } C) = 0 \)[/tex].
Given:
[tex]\[ P(B \text{ AND } C) = 0 \][/tex]
This condition is satisfied, meaning events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are mutually exclusive.
2. Checking if the events are Independent:
Events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are independent if the occurrence of one event does not affect the probability of the other event occurring. This is defined by the condition:
[tex]\[ P(B \mid C) = P(B) \][/tex]
Given:
[tex]\[ P(B \mid C) = 0 \][/tex]
[tex]\[ P(B) = 0.75 \][/tex]
To check for independence, we compare [tex]\( P(B \mid C) \)[/tex] and [tex]\( P(B) \)[/tex]:
[tex]\[ P(B \mid C) = 0 \][/tex]
[tex]\[ P(B) = 0.75 \][/tex]
Since [tex]\( P(B \mid C) \neq P(B) \)[/tex], the events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are not independent.
### Conclusion:
After careful consideration:
- The events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are mutually exclusive.
- The events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are not independent.
Based on these conditions, the correct answer is that events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are Mutually Exclusive.
