 In short can be multiplied by itself times, which gives us

Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is

Applying the Poisson approximation for the binomial on the group of 23 people,

so

The result is over 50% as previous descriptions.This approximation is the same as the one above based on the Taylor expansion that uses formula_17.A good rule of thumb which can be used for mental calculation is the relation

which can also be written as

which works well for probabilities less than or equal to .