NOTE: Was way easier after I realized that "same birthday" meant people who celebrate their birthday on the same day, not people who were born the exact same day AND year. (In case you are wondering, the birthday paradox does not apply to scenario #2... though with a group size of 50, and an assumed average life expectancy of 75 years, turns out there is actually still a ~5% chance that two people were born on the exact same day!)
Cheers! And since the data seems to indicate there's a pretty good chance it's someone's birthday today with a Community this size... happy birthday, whoever you are! :)
Pushed this one pretty hard... Ran 100,000 simulations each for both Leap Years and Non Leap Years, then weighted the Leap Years 97/400 and non Leap Years 303/400 (Based on a prior calculated 400 year Leap Year cycle calculation). Then ran the precision out to 10 decimals! Had fun with this one!! With 100K simulations the curve smooths out pretty nicely.