Question

If \( A \) and \( B \) are two mutually exclusive events with \( P(B) \ne 1 \), then the conditional probability \[ P(A \mid \overline{B}) = \,? \] where \( \overline{B} \) is the complement of \( B \)

• A. \( \frac{1}{P(B)} \)
• B. \( \frac{P(A)}{P(B)} \)
• C. \( \frac{1}{1 - P(B)} \)
• D. \( \frac{P(A)}{1 - P(B)} \)

Hint:

When two events are mutually exclusive, they cannot occur simultaneously. For conditional probability with a complement, apply the definition using \( P(\overline{B}) = 1 - P(B) \).

Answer 1

The Correct Option is D

Given that \( A \) and \( B \) are mutually exclusive, so: \[ A \cap B = \emptyset ⇒ P(A \cap B) = 0 \] Also, \( P(A \cap \overline{B}) = P(A) \), because if \( A \) and \( B \) are disjoint, \( A \subset \overline{B} \).
By the definition of conditional probability: \[ P(A \mid \overline{B}) = \frac{P(A \cap \overline{B})}{P(\overline{B})} = \frac{P(A)}{1 - P(B)} \]