1. Calculate $\mathbb{P}(X=i)$ :
$$
\mathbb{P}(X=i)=\frac{1}{5}, \quad i=1,2,3,4,5
$$
2. Calculate $\mathbb{P}(Y=j \mid X=i)$ :
$$
\mathbb{P}(Y=j \mid X=i)=\frac{1}{i}, \quad j=1,2, \ldots, i
$$
3. Calculate $\mathbb{P}(X=i, Y=j)$ :
$$
\mathbb{P}(X=i, Y=j)=\frac{1}{5 i}, \quad j=1,2, \ldots, i
$$
4. Calculate $\mathbb{P}(Y=5)$ :
$$
\mathbb{P}(Y=5)=\mathbb{P}(X=5, Y=5)=\frac{1}{25}
$$
5. Calculate $\mathbb{P}(X=5 \mid Y=5)$ :
$$
\mathbb{P}(X=5 \mid Y=5)=\frac{\mathbb{P}(X=5, Y=5)}{\mathbb{P}(Y=5)}=\frac{\frac{1}{25}}{\frac{1}{25}}=1
$$
6. Option A is incorrect since $\mathbb{P}(X=5 \mid Y=5) \neq \frac{1}{2}$.
7. Verify $\mathbb{P}(X=5, Y=5)$ :
$$
\mathbb{P}(X=5, Y=5)=\frac{1}{25}
$$
Therefore, option B is correct.
