I have a simple misconception in probability: there are balls arranged in a triangle as such: 1 ball on row 1, 2 balls on row 2, .., n balls on row n. a ball is chosen and I need to find $P(X=i, Y=j)$ where $X$ is the row of the chosen ball, and $Y$ is the index in that row (starting from 1).
I go about it like this: $P(X=i, Y=j) = P(Y=j \mid X=i) \cdot P(X=i) = \frac{1}{i\cdot n}$
But it is incorrect, what is the reason?