Question

If A and B are two events such that $P(B) = \frac{1}{5}$, $P(A | B) = \frac{2}{3}$ and $P(A \cup B) = \frac{3}{5}$, then $P(A)$ is :

• A. $ \frac{10}{15} $
• B. $ \frac{2}{15} $
• C. $ \frac{1}{5} $
• D. $ \frac{8}{15} $

Hint:

Use the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and calculate $P(A \cap B)$ using conditional probability.

Answer 1

The Correct Option is D

We use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] We know that $P(A \cup B) = \frac{3}{5}$, $P(B) = \frac{1}{5}$, and $P(A | B) = \frac{2}{3}$. We can calculate $P(A \cap B)$ as: \[ P(A \cap B) = P(A | B) \times P(B) = \frac{2}{3} \times \frac{1}{5} = \frac{2}{15} \] Now, using the formula: \[ \frac{3}{5} = P(A) + \frac{1}{5} - \frac{2}{15} \] Solving for $P(A)$, we get: \[ P(A) = \frac{8}{15} \] Therefore, the correct answer is $(D)$.