Question

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is: 

• A. \( \frac{3}{5} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{2}{15} \)
• D. \( \frac{2}{5} \)

Hint:

Always start by listing down what is given and identifying which formula connects the required quantity with the given values. Here, finding the intersection first was mandatory.

Answer 1

The Correct Option is D

Concept:
• Definition of conditional probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).
• Addition theorem of probability: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Step 1: Determine the individual probabilities \( P(A) \) and \( P(B) \)
From the given equation \( 3P(A) = P(B) = \frac{3}{5} \):
For \( P(B) \):
\[ P(B) = \frac{3}{5} \]
For \( P(A) \):
\[ 3P(A) = \frac{3}{5} \implies P(A) = \frac{1}{3} \times \frac{3}{5} = \frac{1}{5} \]

Step 2: Calculate the intersection probability \( P(A \cap B) \)
Use the conditional probability formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substitute the known values:
\[ \frac{2}{3} = \frac{P(A \cap B)}{3/5} \]
\[ P(A \cap B) = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5} \]

Step 3: Apply the addition theorem
To find \( P(A \cup B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substitute the values obtained in the previous steps:
\[ P(A \cup B) = \frac{1}{5} + \frac{3}{5} - \frac{2}{5} \]
\[ P(A \cup B) = \frac{1 + 3 - 2}{5} = \frac{2}{5} \]