Why can/can't I find my number in Pi?
If we view Pi as a big, random string of numbers (which is close enough for our purposes), then we can figure out the odds of finding any string in the first 100 million digits of Pi:
Number Length Chance of Finding
1-5 100%
6 Nearly 100%
7 99.995%
8 63%
9 9.5%
10 0.995%%
11 0.09995%
Happily, if you include the zeros, birthdays are 8 digits long -- so you have a 63% chance of finding your birthday in the first 100 million digits of pi. Note that if I added another 100 million digits, these odds would only go up to 86%, so it'll be a while before everyone can find their birthday in Pi.
I've also posted a more in-depth explanation of the probability of finding strings in Pi for people who are curious.