Two well constructed dice are tossed. Let $A$ denote the event that the first die shows $2$ or $5$ and $B$ denote the event that the sum of the values of the numbers shown on the two tossed dice is $\ge7$.
Are the events $A$ and $B$ independent or dependent?
$P(A) = g/m = 2/6 = 1/3$
$P(B) = g/m = 21/36$
$P(A \cap B) = g/m = 7/36$
$P(A)P(B) = (1/3)(21/36) = 7/36$
Since $P(A)P(B) = P(A\cap B)$ we can conclude that the events $A$ and $B$ are independent.
If we would have chosen the numbers $3$ and $5$ instead of $2$ and $5$ in event $A$ then $P(A\cap B) \neq P(A)P(B)$. However, when choosing numbers whose sum equals $7$, like $1$ and $6$, we get that the events are independent.
The math clearly shows that the events are independent when certain numbers are picked, but is there any intuitive way whatsoever of grasping why this is the case?
I find the result very counter intuitive.
Any explanation of this is greatly appreciated!