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



Answer :

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.