 The Birthday Problem

The Proof

We will show that the coincidence, suspected by the author to have no mathematical significance, was not a coincidence at all.

Given: n, the number of days in a year, e.g.,  n= 365; k, the number of partygoers needed for the probability of two identical birthdays to exceed  1 - p, e.g.,  k= 23  for  p= .5; b, the number of different pairings of  k people, that is, b=(k(k-1)/2, e.g., b= 253  for  k= 23; c, the number of partygoers needed for the probability of one of them having  your birthday to exceed  1 - p, e.g.,  c= 253  for  n= 365  and  p= .5; [A] if a<< n [B] if i<< j

we will show that: is not a coincidence, but is true for all reasonable values of  n and  p.  Substitute into [A] yielding:
[C]  Multiply both sides by n
[D]  Raise both sides to the power k
[E]  Simplify the double exponent
[F]  Substitute b for [G]  On the left side, combine the powers of n
[H] [I] [J]  Substitute j = n- 2 , i = k- 2  into  [B] [K] [L]  The right sides of [J] and [L] are equal; pair the left sides
[M]  Multiply both sides by n-1
[N]  Multiply both sides by n
[O] [P]  Divide both sides by [Q] [R]  Substitute p for [S] [T]  Substitute c for [U]  QED. I have not bounded the error analytically, but empirically, I have shown the error in: to be within 1.0 when n> 4 and p> .04.