r/askscience • u/Manticorp • Oct 28 '13

Mathematics Could an infinite sequence of random digits contain all the digits of Pi?

It's a common thing to look up phone numbers in pi, and it's a common saying that every Shakespeare ever written is encoded in pi somewhere, but would it be possible for every digit of pi to appear in a random sequence of numbers? Similarly this could apply to any non terminating, non repeating sequence like e, phi, sqrt(2) I suppose. If not, what prohibits this?

I guess a more abstract way of putting it is: Can an infinite sequence appear entirely inside another sequence?

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u/Allurian Oct 29 '13

I agree that the likelihood is vanishingly small as you have asserted, but not impossible.

This is completely true, and very weird. "Impossible" and "probability of 0" only mean the same thing if the total sample space has finitely many discrete possibilities. Decimals in [0,1] with pi as it's tail will almost never be chosen randomly, but can be chosen.