Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice each of which takes six values $1,2,3,4,5,6$ with equal probability. What is the conditional expectation
\[E\left[X_{1} \mid \max \left(X_{1}, X_{2}\right)=5\right]\]
- $3$
- $4$
- $35 / 9$
- $5 / 2$
- $15 / 4$