The 20th person is used as the starting birthday
The 19th person has 1/365 change of matching that
the 18th person has 2/365 chance of matching the first two
the 17th person has 3/365 chance of matching those three
etc etc etc.
So the sum of the first 19 integers is 19*20/2 = 190
190/365 = 38/73 = 52%
BTW the probability exceeds 92% for 25 or more