A friend of mine has been thinking about the well-known fact that many lists of seemingly random numbers (addresses, physical constants, populations) contain far disproportionately many initial 1's (compared to other initial digits, which decrease in frequency up to 9).
He noted that many physical constants are ratios—constants of proportionality, for example. So he calculated the probability that any two random numbers x and y between 0 and 1 (random with a flat distribution) have a ratio y/x that starts with a 1 (meaning its first non-zero digit: .00013 counts as an initial 1).
Cool answer: 1/3
The proof is left as an exercise (unless there's an outcry for it). Just a little tidbit to tide you over, since I haven't been posting much.