A number $X$ is picked at random from the set $\{ 2,3, \cdots , k \}$. Having picked $X$, another number $Y$ is picked at random from the set $\{1,2, \cdots , X-1 \}$. Find the probability distribution of $Y$.
I can understand that it will be a uniform distribution, but cannot figure out the exact expression.
The hardest part here is the interpretation of the assignment in terms of conditional probabilities. We actually have $$ \mathbb P\left(Y=j\mid X=\ell\right)=\frac1{\ell-1},\quad 1\leqslant j\leqslant \ell-1 \leqslant k-1. $$ Since you also know that $\mathbb P\left(X=\ell\right)=1/(k-1)$, you can find $\mathbb P\left(Y=j, X=\ell\right)$ hence using the law of total probabilities, $\mathbb P\left(Y=j\right)$.