. A popular Dilbert cartoon strip (popular among statisticians, anyway) shows an allegedly
“random” number generator produce 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?