To determine whether events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are independent, mutually exclusive, both, or neither, we need to understand the definitions and apply the given probabilities:
1. Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. In probability terms, this means [tex]\( P(B \cap C) = 0 \)[/tex].
2. Independent Events: Two events are independent if the occurrence of one event does not affect the probability of the other. In probability terms, this means [tex]\( P(B \mid C) = P(B) \)[/tex].
Given:
- [tex]\( P(B) = 0.75 \)[/tex]
- [tex]\( P(B \cap C) = 0 \)[/tex]
- [tex]\( P(C) = 0.55 \)[/tex]
- [tex]\( P(B \mid C) = 0 \)[/tex]
Step-by-step Analysis:
### Check for Mutually Exclusive Events
We know that two events are mutually exclusive if they cannot happen simultaneously, which mathematically means:
[tex]\[ P(B \cap C) = 0 \][/tex]
The information given states:
[tex]\[ P(B \cap C) = 0 \][/tex]
Since [tex]\( P(B \cap C) = 0 \)[/tex], events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are indeed mutually exclusive.
### Check for Independent Events
Two events are independent if:
[tex]\[ P(B \mid C) = P(B) \][/tex]
We are given:
[tex]\[ P(B \mid C) = 0 \][/tex]
To see if events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are independent, we compare [tex]\( P(B \mid C) \)[/tex] with [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = 0.75 \][/tex]
Since [tex]\( P(B \mid C) \neq P(B) \)[/tex] (0 is not equal to 0.75), events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are not independent.
### Conclusion
- Since [tex]\( P(B \cap C) = 0 \)[/tex], events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are mutually exclusive.
- Since [tex]\( P(B \mid C) = 0 \neq P(B) \)[/tex], events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are not independent.
Thus, the correct answer is:
Mutually Exclusive.
