$\begin{align}
\mathbb{E}[X | Y = 5]
& = \sum_{n = 1}^{\infty} n \, \mathbb{P}(X = n | Y = 5)
= \sum_{n = 6}^{\infty} n \cdot \frac{\mathbb{P}(X = n, Y = 5)}{\mathbb{P}(Y = 5)}
= \sum_{n = 6}^{\infty} n \cdot \frac{\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6} \cdot \left( \frac{5}{6} \right)^{n - 6} \cdot \frac{1}{6}}{\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6}} \\
& = \frac{1}{6} \sum_{n = 0}^{\infty} (n + 6) \left(\frac{5}{6}\right)^n
= \ldots
= 11
\end{align}$