File:Birthday paradox.png

Given a room of ${\displaystyle k}$ people, ${\displaystyle P=1-\left(1-{\frac {1}{365}}\right)^{\frac {k(k-1)}{2}}}$ describes the probability that at least two people share the same birthday (assuming ${\displaystyle n=365}$ days per year).

This plot shows that ${\displaystyle P}$ approaches 1 quickly, but does not equal 1 until ${\displaystyle k=366}$