  By a classical result of Gauss and Kuzmin, the continued fraction expansion
of a ``random'' real number contains each digit $a\in\mathbb{N}$ with
asymptotic frequency $\log_2(1+1/(a(a+2)))$.
  We generalize this result in two directions: First, for certain sets
$A\subset\mathbb{N}$, we establish simple explicit formulas for the frequency
with which the continued fraction expansion of a random real number contains a
digit from the set $A$. For example, we show that digits of the form $p-1$,
where $p$ is prime, appear with frequency $\log_2(\pi^2/6)$.
  Second, we obtain a simple formula for the frequency with which a string of
$k$ consecutive digits $a$ appears in the continued fraction expansion of a
random real number. In particular, when $a=1$, this frequency is given by
$|\log_2(1+(-1)^k/F_{k+2})|$, where $F_n$ is the $n$th Fibonacci number.
  Finally, we compare the frequencies predicted by these results with actual
frequencies found among the first 300 million continued fraction digits of
$\pi$, and we provide strong statistical evidence that the continued fraction
expansion of $\pi$ behaves like that of a random real number.
