$X = \{1, 2, 3, ..., n\}$, uniformly destributed, independent from $Y$
$Y = \{1, 2, 3, ..., n\}$, uniformly destributed, independent from $X$
$P(X > Y \cap X = k) = $ ?
Method 1
$P(X > Y \cap X = k)$
$= P(X = k)P(X>Y|X=k)$
$=\frac{1}{n} \cdot \frac{k-1}{n}$
Method 2
$P(X > Y \cap X = k)$
$= P(X>Y)P(X=k|X>Y)$
$= \frac{n-y}{n} \cdot \frac{1}{n-y}$ ?
Method 1 looks correct
Method 2 introduces an undefined $y$ which you should probably try to avoid, perhaps something like
$P(X > Y \cap X = k)$
$= P(X>Y)\,P(X=k\mid X>Y)$
$= \displaystyle \sum_{y=1}^n P(Y=y)\,P(X>Y\mid Y=y )\,P(X=k\mid X>Y, Y=y)$
$= \displaystyle \sum_{y=1}^n \frac1n \cdot \frac{n-y}{n} \cdot \frac{I[k \gt y]}{n-y} $ using an indicator function
$= \displaystyle \sum_{y=1}^{k-1} \frac{1}{n^2} $
$= \displaystyle \frac{k-1}{n^2}$