Birthdays

Let Pr(A)= the probability that at least one person is born on Jan. 01.

Let Pr(~A)= the probability that no one is born on Jan. 01.

It is clear that Pr(A)=1-Pr(~A) => (~ means not: all of something plus everything else must equal everything: A plus everything that isn't A must equal everything)

So now all we must do is find Pr(~A)

The probability that one person is not born on Jan. 01 is 364/365. Therefore, the probability that 253 people are not born on Jan. 01 is (364/365)^253

(364/365)^253 = Pr(~A) = 1-Pr(A) = 0.49952

1-Pr(A)=0.49952

1-0.49952=Pr(A)

0.5005=Pr(A)

It would not be 253/365 because some of the people could share a birthday that is not January 1st, thereby decreasing the chance. Using binomial theorem: $\sum _{ x=1 }^{ 253 }{ { _{ 253 }{ C }_{ x } } } { \left( \frac { 1 }{ 365 } \right) }^{ x }{ \left( \frac { 364 }{ 365 } \right) }^{ 253-x }$