On the decimal and octal digits of $1\mathrm{/}p$

Let $p$ be a prime $\equiv 3$ mod 4, $p>3$, and suppose that 10 has the order $(p-1)\mathrm{/}2$ mod p. Then $1\mathrm{/}p$ has a decimal period of length $(p-1)\mathrm{/}2$. We express the frequency of each digit $0,\dots ,9$ in this period in terms of the class numbers of two imaginary quadratic number fields. We also exhibit certain analogues of this result, so for the case that 10 is a primitive root mod $p$ and for the octal digits of $1\mathrm{/}p$.