I am solving another problem and it boils down to the following.
Say we have a bag with identical balls numbered 1,2,...,100 and we draw 5 of them on each attempt.
Say we make N draws/attempts and after each draw we put back the 5 drawn ones and we shuffle the bag.
Event A = {Ball i was drawn at least once after the N draws}
Event B = {Ball j was drawn at least once after the N draws}
Here $i \ne j$
Are these two events independent?
My intuition tells me they are but I am not absolutely sure and somehow I fail to convince myself. But who knows, maybe my intuition is misleading me here.
Two events $A,B$ are said to be independent if and only if $P(A\cap B) = P(A)P(B)$. Suppose $N=1$. Then:
$$P(A) = \dfrac{\dbinom{99}{4}}{\dbinom{100}{5}} = \dfrac{1}{20} \\ P(B) = \dfrac{\dbinom{99}{4}}{\dbinom{100}{5}} = \dfrac{1}{20} \\ P(A\cap B) = \dfrac{\dbinom{2}{2}\dbinom{98}{3}}{\dbinom{100}{5}} = \dfrac{1}{495} \neq P(A)P(B)$$
The two events you list are not independent.