Thanks Eric, for your contribution.
My comments were already formulated (the last part may be the paradox you mentioned):
To start with, the heated discussions were about the precise meaning of the question. Was or wasn't it allowed to have a date with more than 2 anniversaries? Was or wasn't it allowed to have (more than) 2 couples of anniversaries? To my opinion the question means in normal every day language that we are interested in knowing the chances of an occurrence that at least two people celebrate on the same day. I came to that conclusion (and to a computable answer) by reversing the question: how about each and every of the lot celebrating on different days?
That's very easy to answer by the definition of 'chance':
# wanted situations / # possible situations.
# wanted situations = 365x364x363...x340 (365!/(36526)!)
# possible situations = 365x365x365...x365 (365**26)
(of course you'll have to change 365 by 366 for a leap year, but that doesn't make much difference)
With for example Free42 (a real HP42S is slightly too "limited") you can easily calculate the quotient directly, but otherwise you'll have to program (almost any HP calc will do).
Chance=1 ; FOR i=0 TO 25 ; Chance=Chance*(365i)/365 ; NEXT
which may translate for the HP41C(V) into this:
*LBL CHANCE
1

1000
/
STO 00
1
STO 01
*LBL 01
365
RCL 00
INT

ST* 01
365
ST/ 01
ISG 00
GTO 01
RCL 01
26 XEQ CHANCE => 0.40(17591805)
The answer 0.40 is the answer for the reversed question. So the answer to the original question is 10.40 => 0.60.
So take 26 people at random, for example 26 readers of the HP museum forum and there's 60% chance that at least 2 of them were born on the same day in the year.
And that's an unexpectedly high value, isn't it?
With an increasing number of people the chance for 'all different dates' will approach zero and hence the chance for 'a duplicate' will approach certainty. With 366 or more people a duplicate will have become certain.
Strangely enough, when I would ask (the other way round) for the chance for the existence of a date on which nobody has his/her birthday then this chance will never be equal to zero, no matter how many people I put together. Hard to believe that even for the entire Dutch population (about 16 million persons) there's a chance, albeit extremely low, that on one (or more!) date(s) nobody celebrates his/her birthday. It will never happen, but there is a chance ...