A ball is drawn at random from a box containing 10 balls numbered sequentially from 1 to 10. Let X be the number of the ball selected, let R be the event that X is an even number, let S be the event that $\displaystyle X\geq 6$, and let T be the event that $\displaystyle X\leq 4$. Which of the pairs (R,S), (R,T), and (S,T) are independent?I can solve this mathematically as such:

$\displaystyle \newline P(R)=0.5\newline P(S)=0.5\newline P(T)=0.4$

$\displaystyle P(R\cap S)=P\{6,8,10\}=0.3\neq P(R)*P(S)=0.25$

$\displaystyle P(R\cap T)=P\{2,4\}=0.2= P(R)*P(T)=0.20$

$\displaystyle P(S\cap T)=P\{\emptyset\}=0\neq P(S)*P(T)=0.20$

So $\displaystyle P(R\cap T)$ is independent.

My question is, intuitively, how is $\displaystyle P(R\cap T)$ independent, but $\displaystyle P(R \cap S)$ not? Maybe I'm not understanding what to events being independent entails.