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The difficulty of picking numbers at random

Calculate the exact percentage, within a randomly selected list, of neighbouring digits that would have the same value, and the percentage of neighbours whose value differs by 1 (and do NOT count 0 and 9 as differing by 1). What is the probability of a list of 40 randomly choosen digits containing no same value neighbours? Have Mathematica select 40 digits at random, and ask a friend not in this class to do the same. Do both these lists match your expectations for randomly selected lists?


Solution Preview

The probability to have a decimal digits sequence of length N with no same value neighbours is

p_0(N) = (9/10)^{N-1}

In the case of N = 40, we have

p_0(40) ≈ ...

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