Question

If A and B are events such that P(A)=\(\frac{1}{4}\), P(A/B)=\(\frac{1}{2}\) and P(B/A)=\(\frac{2}{3}\) then P(B) is

• A. \(\frac{2}{3}\)
• B. \(\frac{1}{6}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{1}{3}\)

Hint:

When given conditional probabilities, use the definitions of conditional probability to relate the joint probability \( P(A \cap B) \) with the individual probabilities. Solve for the unknown probability using these relationships.

Answer 1

The Correct Option is D

The correct answer is: (D): \( \frac{1}{3} \)

We are given the following probabilities:

\[ P(A) = \frac{1}{4}, \quad P(A | B) = \frac{1}{2}, \quad P(B | A) = \frac{2}{3} \]

We are tasked with finding \( P(B) \).

Step 1: Use the definition of conditional probability
Recall the definition of conditional probability:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

Substitute \( P(A | B) = \frac{1}{2} \) into the equation:

\[ \frac{P(A \cap B)}{P(B)} = \frac{1}{2} \]

From this, we can solve for \( P(A \cap B) \) in terms of \( P(B) \):

\[ P(A \cap B) = \frac{1}{2} P(B) \]

Step 2: Use the definition of \( P(B | A) \)
Similarly, from the definition of conditional probability for \( P(B | A) \), we have:

\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]

Substitute \( P(B | A) = \frac{2}{3} \) and \( P(A) = \frac{1}{4} \) into the equation:

\[ \frac{P(A \cap B)}{\frac{1}{4}} = \frac{2}{3} \]

This simplifies to:

\[ P(A \cap B) = \frac{2}{3} \times \frac{1}{4} = \frac{1}{6} \]

Step 3: Solve for \( P(B) \)
From Step 1, we know that \( P(A \cap B) = \frac{1}{2} P(B) \). Now substitute \( P(A \cap B) = \frac{1}{6} \) into this equation:

\[ \frac{1}{2} P(B) = \frac{1}{6} \]

Solving for \( P(B) \), we get:

\[ P(B) = \frac{1}{3} \]

Conclusion:
The probability of event \( B \) is \[ \frac{1}{3} \] so the correct answer is (D): \( \frac{1}{3} \).