A popular Dilbert cartoon strip (popular among statisticians, anyway) shows an allegedly “random” number generator producing the sequence 999999 with the accompanying comment, “That’s the problem with randomness: you can never be sure.” Most people would agree that 999999 seems less “random” than, say, 703928, but in what sense is that true? Imagine we randomly generate a six-digit number; i.e., we make six draws with replacement from the digits 0 through 9.

a. What is the probability of generating 999999?
b. What is the probability of generating 703928?
c. What is the probability of generating a sequence of six identical digits?
d. What is the probability of generating a sequence with no identical digits? (Comparing the answers to (c) and (d) gives some sense of why some sequences feel intuitively more random than others.)
e. Here’s a real challenge: what is the probability of generating a sequence with exactly one repeated digit?