Question

If \[ P(A)=0.5, \qquad P(B)=0.3, \qquad P(A\cap B)=0.1, \] find \[ P(A|B). \]

• A. \(0.1\)
• B. \(0.2\)
• C. \(\frac13\)
• D. \(0.5\)

Hint:

Conditional probability is always calculated as \[ \frac{\text{Intersection}}{\text{Given Event}} \] provided the denominator is non-zero.

Answer 1

The Correct Option is C

Concept: Conditional probability measures the probability of occurrence of one event when another event has already occurred. The formula is \[ P(A|B) = \frac{P(A\cap B)} {P(B)} \] provided \[ P(B)\neq0. \]

Step 1: Write the given values \[ P(A)=0.5 \] \[ P(B)=0.3 \] \[ P(A\cap B)=0.1 \]

Step 2: Substitute into the conditional probability formula \[ P(A|B) = \frac{0.1}{0.3} \] \[ = \frac{1}{3} \]

Step 3: Verify the result Since probability lies between 0 and 1, \[ \frac13 \] is a valid probability value. Final Answer: \[ \boxed{\frac13} \]