Expected smallest roll seen before the first five

A fair die is rolled repeatedly, stopping as soon as a $5$ appears. Taking all rolls into account, including the final $5$, what is the expected value of the smallest number rolled?

$\tfrac{137}{60}$. The minimum is $m<5$ exactly when $m$ appears before the first $5$, which in turn comes before every value below $m$. By symmetry among first appearances this has probability $\tfrac{(m-1)!}{(m+1)!}=\tfrac{1}{m(m+1)}$, and the minimum is $5$ with probability $\tfrac15$. So $\mathbb{E}=\sum_{m=1}^4\tfrac{1}{m+1}+1=\tfrac{137}{60}$.