your chance of winning with N other people = N/365 [you match someone] * (365!/((365-N)! * 365^N)) [no other matches]
Using this table of results for the latter expression (
link):
N 365*P (for calculation ease)
18 11.754
19 11.799
20 11.780
21 11.676
This implies that you should want 19 people in front of you, which will allows you to win 3.23% of the time. This seems low, but I think it is right nonetheless.
You people should be ashamed of yourselves for Monte-Carloing this.
Note: the correct answer has nothing to do with when there's a 50% likelihood of someone having won before you. P(someone else winning) increases in an S-curve shape, while P(you winning, assuming no one else has won) increases linearly. The idea is to take advantage of as much of the linear increase as you can before the steep portion of the S starts killing your equity.