system
Given the following post, respond as if you're commenting on Reddit:
human
Birthday Paradox

This problem is related to the birthday paradox but I could not  find any calculator online so now I'm asking you. 

If there are 2700 people in a room, what is the probability that 81 of them are born on the same date. If you could show me a formula you used that would be great too!
gpt
There are 2700!/(2700-81)!81!&nbsp;=&nbsp;N&nbsp;≅&nbsp;10^(156.65) groups of 81 in 2700 people. In each group there is a (1/365)^(80)&nbsp;=&nbsp;ε&nbsp;≅&nbsp;10^(-204.98) probability that everyone shares the same birthday, so the probability of _not_ having the same birthday is 1-ε. The probability of having N groups _not_ sharing a birthday is p&nbsp;=&nbsp;(1-ε)^(N) (this last expression can be approximated with 1-εN for ε much smaller than 1/N). The probability of having _at least_ one group with a shared birthday is thus 1-(1-ε)^(N)&nbsp;≅&nbsp;εN&nbsp;≅&nbsp;10^(-48.33). That's around one in the number of atoms in the Earth.