Die is rolled once. Are events of "result is even" and result is greater than 3 independent?

Question

Are the events of a die result being even or >3 independent?

I can't find the exact proof online and was wondering if this would be the correct solution. I know that for independent events $P(A|B)=P(A)$ but for a case of a die, $P(even)=P ${2,4,6}$=1/2$ $P(>3)=P${4,5,6}$=1/2$ Now since $P(even|>3)=P {4,6}=2/3 \ne 1/2$, and $P(>3|even)=2/3 \ne 1/2$ does that mean they are not independent? -> therefore they are dependent?

Additionally for independent events : $P(A \cap B)= P(A) × P(B)$ but $P(even)=1/2$ and $P(>3)=1/2$, but $P(even \cap >3)= 2/6 \ne1/4$

is this correct reasoning? thanks!

Answer 1

Yes, both lines of reasoning are correct!

And just intuitively: If I roll a die and tell you that the outcome is greater than $3$, then you know there is a more than $50$% chance it is even, which is what the chance would be if I didn't tell you anything ... so these events are positively correlated, i.e. not independent.