, 2023

Solution:

For each $i = 1, 2, \ldots, 6$, let $X_{i}$ denote the indicator variable of whether the $i$-th die shows the same number as the sixth die. Clearly, $X_{6} = 1$ always. For all other $i$, $X_{i}$ is $1$ with probability $\frac{1}{6}$ and $0$ otherwise, so $\mathbb{E}\left[X_{i}\right] = \frac{1}{6}$.

By linearity of expectation, the answer is
$$\mathbb{E}\left[X_{1} + \cdots + X_{6}\right] = \mathbb{E}\left[X_{1}\right] + \cdots + \mathbb{E}\left[X_{6}\right] = 5 \cdot \frac{1}{6} + 1 = \frac{11}{6}$$