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: Rearrange the addition rule to isolate the intersection:
$P(A\cap B)=P(A)+P(B)-P(A\cup B)$.

Step 2: Substitute with a common denominator of 6:
$P(A)=2/6,\ P(B)=3/6,\ P(A\cup B)=4/6$. So $P(A\cap B)=2/6+3/6-4/6=1/6$.

Step 3: Independently compute the product $P(A)P(B)$ and compare:
$P(A)P(B)=(1/3)(1/2)=1/6$, matching $P(A\cap B)$ exactly.

Final Answer:
The equality $P(A\cap B)=P(A)P(B)=1/6$ confirms independence. \[ \boxed{A,B\text{ independent}} \]