I just sat down this morning and decided to see if I could correct the
previous birthday match formula that I had posted for a world with an integer
number of days to more properly account for leap year effects than simple
interpolation on the integer formula. The result was surprisingly
straightforward to calculate--again with some simplifying assumptions such as
(1) the distribution of birthtimes being strictly uniform throughout an actual
year, and (2) that the date and time for which the experiment is performed is
chosen by randomly picking an instant throughout that part of history since
the Gregorian Calendar came into use (so as to prevent anomalous correlations
in the probability of a leap date birthday due to the finite longevity in
the lifetimes of the participants and the current proximity of the next
leap year). With these assumptions the probability for there being at least
one birthday match in a group of n people on a planet with a year of d + r
days where d is a nonnegative integer and r is a remainder fractional day
value between 0 and 1 is:
match prob. = 1 - (d!)*(d + 1 - n*(1 - r))/(((d - n + 1)!)*(d + r)^n)
For our planet we have d = 365 and r = 0.2425 if the Gregorian year is used.
I leave it to the reader to determine the probabilities for the n =365 and
366 cases. Maybe an industrious reader can compare the n = 50 result
given with by formula with that of the integer formula using both the
d = 365 case, and with the interpolative d = 365.2425 case.
I hope that Brian does not overly object to this formula. Remember, any
such formula requires that at least some assumptions are made. If you do
not like the assumptions, don't blame the formula.