I have the following from Joseph Blitzstein's Introduction to probability:
Let $X \sim DUnif(C)$, and $B$ be a nonempty subset of $C$. Find the conditional distribution of $X$, given that $X$ is in $B$.
Following is my attempts:
$\begin{align} P(X=k\mid X\in B) &= \frac{P(X\in B|X=k)P(X=k)}{P(X\in B)}\\ &= \frac{[P(X\in B|k\in B, X=k)P(k\in B|X=k) + P(X\in B| k\notin B, X=k)P(k\notin B|X=k)]P(X=k)}{P(X\in B)}\\ &=\frac{P(k \in B| X=k)P(X=k)}{P(X\in B)}\\ &=\frac{P(X=k|k\in B)P(k\in B)}{P(X\in B)} \end{align}$
However, I am stuck on how to interpret $P(k\in B)$ since $k$ is a free variable if my understanding of the definition of a PMF is correct.
After looking online for solutions, I stumbled on this old post What is conditional distribution? and it seems like I used an approach similar to one of the answers. However, in that post, it is stated that $P(X=k\mid X\in B) = 1/|B|$, which makes me wonder if in my case, $P(k\in B) = P(X\in B)$ and $P(X=k|k\in B) = 1/|B|$.
Any help will be appreciated.
If $X$ is uniformly distributed over $C$, then the probability that $X$ lies in $B\subset C$ is the relative cardinality $|B|/|C|$. In particular, $P(X=k) = 1/|C|$. And also $P(X=k, X\in B) = P(X=k)$ or $0$, according as $k\in B$ or not. From this you'll be able to work out $P(X=k\mid X\in B)$.