[omra]

Assuming the digits of pi are randomly distributed, any finite digit sequence can be found in pi.

The probability any sequence of length d is found in N digits of pi is 1 - 1/exp(N*0.1^d) (Poisson distribution for approximating the binomial). Then the limit as N approaches infinity is 1 for any finite d.

[Someone]

Amusing fact: the probability that a sequence can be found is not equal for all sequences.

A simple example: 2 binary digits in binary sequences of length 3. There are eight binary sequences of length 3:

  000
  001
  010
  011
  100
  101
  110
  111

3 of those contain '00' but 4 of them '01'. Reason for the discrepancy is that one of those with '00' has 2 overlapping occurrences, but is counted only once. You get this as soon as overlap can occur, i.e. when the sequence to be found starts with x digits that it also ends with.

Of course, none of this matters, especially not when d << N, which it will be if N goes to infinity.

Also, the mathematical term is 'normal number' (http://mathworld.wolfram.com/NormalNumber.html), and we do not know whether pi is normal.

[hypersoar]

The probability of an event being 1 is _not_ the same thing as that event being completely certain. For example, if you pick a random real number between 0 and 1, the probability of getting something rational is zero. It's clearly not impossible, though.

[gizmo686]

Do you know of any proof for that? My math intuition is telling me that randomly picking a real number is guarenteed to be irrational, based on the fact that there is an uncountable infinity real numbers, but only a countable infinity of rational numbers. But, without assuming a probability of 0 means impossible, I do not know how to go about proving/disproving this.

[hypersoar]

We can't really pick "random real numbers" in any practical sense, so this is pretty much a theoretical distinction. It's essentially a matter of definition. The rational numbers have probability zero of being drawn, but they still lie in the sample space.

One way to see it is this: The probability of picking any particular point in the interval is 0. But that doesn't mean that picking that point is impossible. _Some_ point has to show up when you pick one at random.

[philh]

(The digits of pi are definitely not randomly distributed, since they can be generated by a deterministic algorithm; and a randomly distributed infinite digit sequence need not have that specific probability of finding a d-length sequence in the first N digits, unless the random distribution is specifically uniform. Someone is correct that the relevant term here is 'normal'.)