Coming Up Through the Ranks

Through some special features at Amazon Author Central, I am able to know the daily sales rank of Math Jokes 4 Mathy Folks. My sales rank at the end of each of the last three days was 44,404, 96,990, and 35,355, respectively. I thought that was interesting — three days in a row when the sales rank was a five-digit number in which one digit occurred at least three times. What’s the likelihood of that? Stated more formally:

Assuming that the sales rank of MJ4MF is always a five-digit number, what is the probability that three consecutive days’ sales ranks will contain a digit that occurs in the sales rank at least three times?

The sales rank of MJ4MF has never been a five-digit number in which the same digit is repeated five times. (Bummer!) The probability of that occurrence, though, is even less likely than the situation described above — though I won’t tell you exactly how much less likely, so as not to spoil your fun!