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 identify the relationship between the intersection, union, and conditional probability first.
Double check fractional additions by using a common denominator.

Answer 1

The Correct Option is D

Concept:
Conditional probability is given by: \[ P(A|B)=\frac{P(A\cap B)}{P(B)} \] The addition theorem of probability is: \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] 
Step 1: Find \(P(A)\) and \(P(B)\)
Given: \[ 3P(A)=\frac{3}{5} \] Therefore, \[ P(A)=\frac{1}{5} \] Also, \[ P(B)=\frac{3}{5} \] 
Step 2: Find \(P(A\cap B)\)
Using conditional probability: \[ P(A\cap B)=P(A|B)\times P(B) \] Given \(P(A|B)=\frac{2}{3}\): \[ P(A\cap B)=\frac{2}{3}\times\frac{3}{5} \] \[ P(A\cap B)=\frac{2}{5} \] 
Step 3: Calculate \(P(A\cup B)\)
Using the addition theorem: \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] \[ =\frac{1}{5}+\frac{3}{5}-\frac{2}{5} \] \[ =\frac{2}{5} \] 
Final Answer:
According to the supplied values: \[ \boxed{P(A\cup B)=\frac{2}{5}} \] Hence, the given answer corresponds to Option (D).