Question

question
given the following information, determine whether events b and c are independent, mutually exclusive, both, or neither.

• ( p(b) = 0.6 )
• ( p(b \text{ and } c) = 0 )
• ( p(c) = 0.4 )
• ( p(b|c) = 0 )

select the correct answer below:
○ independent
○ mutually exclusive
○ both independent & mutually exclusive
○ neither

Explanation:

Step1: Recall definitions

• Mutually Exclusive: Events \( B \) and \( C \) are mutually exclusive if \( P(B \text{ AND } C) = 0 \) (they can't occur at the same time).
• Independent: Events \( B \) and \( C \) are independent if \( P(B \text{ AND } C) = P(B) \times P(C) \).

Step2: Check Mutual Exclusivity

Given \( P(B \text{ AND } C) = 0 \), this satisfies the condition for mutually exclusive events.

Step3: Check Independence

Calculate \( P(B) \times P(C) \):
\( P(B) \times P(C) = 0.6 \times 0.4 = 0.24 \).
But \( P(B \text{ AND } C) = 0
eq 0.24 \), so they are not independent.

Answer:

Mutually Exclusive