Independence of drawing a labelled colored ball A bag contains $5$ red balls and $5$ blue balls. The red balls are labelled $1\cdots5$ and the blue balls are labelled similarly. Let $\text{A}$ denote the event "Ball drawn is red", and $\text{B}$ denote the event "Ball is labelled by $2$". Are $\text{A}$ and $\text{B}$ independent events? Please clarify.

$P(A) = {5\over10} = {1\over 2}$

$P(B) = {2\over10} = {1\over5}$

$P(A)\cdot P(B) ={1\over 10}$

$P(A\cap B) = {1\over 10}$

Since $P(A)\cdot P(B) = P(A\cap B)$, the mathematical definition of independence is satisfied, so events $A$ and $B$ are independent