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Given the following information, determine whether events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are independent, mutually exclusive, both, or neither.

- [tex]\( P(B)=0.75 \)[/tex]
- [tex]\( P(B \text{ AND } C)=0 \)[/tex]
- [tex]\( P(C)=0.55 \)[/tex]
- [tex]\( P(B \mid C)=0 \)[/tex]

Select the correct answer below:

A. Mutually Exclusive
B. Independent
C. Neither
D. Both Independent & Mutually Exclusive


Sagot :

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.