Sequences of digits and PI?

The probability of finding any digit string can be readily calculated providing the didgits of pi are random. Empirical evidence seems to support the randomness. The deviation from uniform digit distribution in the first trillion digits is no more than 10^-6 of course, this is no proof of randomness and the randomness of the digits of pi is a hot topic of discussion.

If things were random, you would expect any m digit string to show up n times in the first n x 10^m digits of pi. There is a tabulation of all instances 8 repeated digits (e.g., 33333333) in the first 2 x 10^8 digits. You would expect 20 such occurances based on random chance and there are 20 such occurrances. There are 3 occurrances of 9 digits compared to 2 expected.

Based on this, it seems very likely that the digits are random and therfore any decimal sequence of n digits should occur about once in 10^n places of pi. Of course "likely" has no place in math. There is a famous example where a "likely" conjecture held true for the first 10^18 integers and then failed.

All the data for these wild allegations comes from my all-time favorite math site, MathWorld. Specifiaclly:

http://mathworld.wolfram.com/PiDigits.html

That is an extremely interesting idea!

I think the issue at hand is that pi is infinite. The probability of a sequence 123456789 or 1111111111 existing in a really long decimal would be so minute that it is essentially considered nonexistant. However, since pi is of infinite length, I believe that such a sequence would occur...well, eventually. (This is like saying that if you flip a coin 10000 times, eventually you'll get 4 heads in a row...except on a much larger, infinite scale.)

The given sequence of digits most probably does occur in the decimal representation of pi but I can't see why it must. As for any specified sequence how can we ever know until it is actually found? It's rather like the reverse of proving or disproving a conjecture. One example that doesn't work disproves it, but many of examples working will not prove it in general. In this case one example of it found proves it correct but any number of digits examined without it being found does not prove that it can't be there.

there is no reason why all possible 10 digit sequences should appear in the decimal representation .. or why they shouldn't.

all you do know is that there is not some fixed length sequence that repeats indefinately from then on.

you know that its perfectly okay for strings to be repeatedly contained within it. for example the strings of length 1 = digits have to keep repeating, just not in a fixed order

Try this website

http://www.angio.net/pi/bigpi.cgi

It will search for digits in Pi. Try your sequence there!