How often does the number 31 recur in an infinite random digit string?

If an infinite digit string is random, then a substring that might occur once in a billion digits, would have probability one of occurring infinitely many times. In fact, there is a finite probability for any finite substring, implying probability one that all finite substrings will recur infinitely often.

The other side of the coin is that probability one doesn't mean absolute certainty. For example, there is probability one that Goldbach's conjecture is true, and yet there might be some case over infinity where it doesn't hold.

Additionally, nearly all the reals are noncomputable. But there is a notional possibility that a a program that generates a 0 or 1 randomly over infinity might generate such a noncomputable. There is no reason to suspect that every noncomputable has the property of every finite substring infinitely recurring. Let us assume such a string. We then scan the string and delete the number 31 in every occurrence. The string is still infinitely long -- but someone might object that the modified string is not now random.

BUT, we have proved such a number exists, a number that might have been generated by a random process, whatever the probabilities say.

A mathematician comments: The answer lies in an explicit definition of the term "infinite random digit string." Any algorithm for producing the digit string will at once tell you your answer and also raise the objection that the digit string isn't really random. A physical system for producing the digits (say, from output of a device that measures quantum noise) turns this from a mathematical into an experimental question. [1]

Conant replies: Or a philosophical question. I.e., we know, according to Cantorian set theory -- as modified by Frankel, Zermelo, Von Neumann (and including the axiom of choice) -- that there is a nondenumerable infinity of noncomputables. So one of these noncomputables R0 -- that could only notionally be reached by some randomization method such as via quantum noise -- does contain a (countable) infinity of 31s. But that means there is also a number R1 that contains the same infinite string as R0 but that lacks all occurrences of 31.

Of course, it isn't necessary that such a randomization process yield a noncomputable. R0 might be a noncomputable and R1 a computable. Nevetheless such a procedure might yield a noncomputable R1.

If we regard Frankel-Zermelo set theory as our foundation, it is pretty hard to imagine an element of the noncomputables as anything but a random number. However, if one could randomly select each digit via a quantum noise detector, we would agree that the probability is overwhelming that '31' will recur an infinity of times.

[1] Private communication with a professor 'pen pal.'