Question: For any pair , where is an irrational number such that , so , and is a natural number, so , what's the probability that the finite string of digits in will be contained in the decimal expansion of ? In other words, what's the probability that such that for all ?
Extra Questions: 1.) What's the probability that the string of digits in occurs in twice? times? Infinitely many times?
2.) How does the answer change (if at all) if we allow to be any real number?
Example: Say and . Then you have so the string of digits of does occur in .