Question

If \(P(A)=\dfrac13\), \(P(B)=\dfrac12\) and \(P(A\cup B)=\dfrac23\), then show that \(A\) and \(B\) are independent events.

Hint:

Find P(A intersect B) from the addition rule, then check if it equals P(A) times P(B).

Answer 1

Step 1: Find \(P(A\cap B)\) using the addition rule:
\[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \]
\[ \dfrac23=\dfrac13+\dfrac12-P(A\cap B) \implies P(A\cap B)=\dfrac13+\dfrac12-\dfrac23=\dfrac{2+3-4}6=\dfrac16 \]

Step 2: Compute \(P(A)\cdot P(B)\):
\[ P(A)\cdot P(B)=\dfrac13\times\dfrac12=\dfrac16 \]

Step 3: Compare:
\(P(A\cap B)=\dfrac16=P(A)\cdot P(B)\), which is exactly the independence condition.

Final Answer:
Since \(P(A\cap B)=P(A)P(B)\), \(A\) and \(B\) are independent. \[ \boxed{P(A\cap B)=P(A)P(B)=\dfrac16} \]