Question

For two events A and B, let $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{4}$ and $P(A \cup B)=\frac{1}{2}$ then $P(A/B)$ is

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{1}{12} \)
• D. \( \frac{1}{4} \)

Hint:

Always remember the fundamental probability formulas: 1. Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. This is crucial for finding the intersection probability when union is given. 2. Conditional Probability: $P(A/B) = \frac{P(A \cap B)}{P(B)}$. This defines the probability of event A occurring given that event B has already occurred. These two formulas are frequently used together in problems involving conditional probability and unions/intersections of events.

Answer 1

The Correct Option is B

We are given the following probabilities: $P(A) = \frac{1}{3}$ $P(B) = \frac{1}{4}$ $P(A \cup B) = \frac{1}{2}$ We need to find $P(A/B)$, which is the conditional probability of A given B. The formula for conditional probability is: $$ P(A/B) = \frac{P(A \cap B)}{P(B)} $$ To use this formula, we first need to find $P(A \cap B)$. We can use the formula for the probability of the union of two events: $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
Step 1: Calculate $P(A \cap B)$. Rearrange the union formula to solve for $P(A \cap B)$: $$ P(A \cap B) = P(A) + P(B) - P(A \cup B) $$ Substitute the given values: $$ P(A \cap B) = \frac{1}{3} + \frac{1}{4} - \frac{1}{2} $$ To add and subtract these fractions, find a common denominator, which is 12. $$ P(A \cap B) = \frac{4}{12} + \frac{3}{12} - \frac{6}{12} $$ $$ P(A \cap B) = \frac{4 + 3 - 6}{12} = \frac{7 - 6}{12} = \frac{1}{12} $$
Step 2: Calculate $P(A/B)$. Now use the conditional probability formula with the calculated $P(A \cap B)$: $$ P(A/B) = \frac{P(A \cap B)}{P(B)} $$ Substitute the values: $$ P(A/B) = \frac{\frac{1}{12}}{\frac{1}{4}} $$ $$ P(A/B) = \frac{1}{12} \times \frac{4}{1} $$ $$ P(A/B) = \frac{4}{12} = \frac{1}{3} $$