Q:

Independent versus dependent event

Problem 1: Let $A$ be the event of getting dealt a flush. Let $B$ be the event of getting dealt a hand in which at least two cards have the same face value (2-10, jack, queen, king, ace). Are $A$ and $B$ independent?
Problem 2: Roll two dice, and let $A$ be the event that the dice match, and let $B$ be the event that the dice sum to $8$. Are $A$ and $B$ independent?
I found that $P(A|B) \not= P(A)$ for both, but I'm not sure that I did my calculations correctly.
$P(A|B) = P(0.00198*1)/P((13/52)*(12/51)$
$P(A|B) = P((6/36)*(1/6))/P(5/36)$

A:

First problem: We have $\Pr(A)\ne 0$ and $\Pr(B)\ne 0$. But clearly $\Pr(A\cap B)=0$. 
Thus $\Pr(A\cap B)\ne \Pr(A)\Pr(B)$. It follows that $A$ and $B$ are not independent.
Second problem: We have $\Pr(A)=\frac{1}{6}$ and $\Pr(B)=\frac{5}{36}$.
The event $A\cap B$ occurs precisely if we get double-$4$ on the dice. This has probability $\frac{1}{36}$,
Note that we have $\Pr(A\cap B)\ne\Pr(A)\Pr(B)$, so the events are not independent.

