-Here’s a real challenge: what is the probability of generating a sequence with

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-Here’s a real challenge: what is the probability of generating a sequence with exactly one repeated digit?

46. 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 Here's a real challenge: what is the probability of generating a sequence with exactly one repeated digit? (e)

Explanation / Answer

e)
6 digit number can be formed using digits 0 to 9 in 10^6 ways (i.e. each digit can be selected in 10C1 ways and there are 6 such selections, hence 10C1*10C1*10C1*10C1*10C1*10C1 = 10^6)

The number of sequences with exactly one repeated digit,
In 10C1 ways we can select a digit which needs to be repeated.
Remaining 4 digits can be selected in 9C4 ways.
These 6 digits can be arranged in 6!/2! ways (dividing by 2! as 1 digi is reapeated)

Hence possible number of sequences are 10*9C4*6!/2! = 453600

Required probability = 453600/10^6 = 0.4536

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