Quote:
Originally Posted by EdMan
I'm not sure that having 367 people in a room guarantees that at least two of them will have the same birthday.

Here's how it works.
1. The only requirement is that two people in the group share the same birthday.
It doesn't have to be any particular day and they don't have to share it with you, just with someone else.
2. The first person in the group has a birthday, for example, 1st January. If the second person in the group is also born on 1st January then, boom, two people share a birthday and the set of two is sufficient. However, the odds of this are very low:1 chance in 365.25 or about 0.3%. (The extra .25 is for 29 Feb, which only happens every four years, of course.)
3. However, if the first person has a birthday on 1 Jan and the second has a different day (say 2 Jan) then close but no cigar. So bring in a third person. If that person has a birthday on either 1 or 2 Jan, then boom. Now the odds are actually 0.8%  I'll put the maths below.
4. If we keep repeating the process, by the time you get to 23 people it's 50% likely that two share a birthday. That's the paradox and, again, see below.
5.
However, we want the odds to be 100%. So the first person in the group has a birthday on 1 Jan; the second person on 2 Jan, the third on 3 Jan... the 31st on 31 Jan, the 32nd on 1 Feb... the 60th on 29 Feb (that rare leap baby), the 61st on 1 Mar... and the 366th on 31 Dec. Whatever birthday the 367th has, they will share it with someone else in the group.
The maths:
 For person 1, the chances are 100% because every date is clear. For person two, there's one day they would share with person 1, but the other 364 are clear, so their chance of a unique birthday is 364/365. For person 3 it's 363/365, and so on through to person 23, whose probability of having a unique birthday is 343/365.
 To find the probability of everyone in the group having unique birthdays, we multiply all those 23 probabilities together, and if we do we end up with a probability of 0.491.
 The probability that a birthday is shared is therefore 1  0.491, which comes to 0.509, or 50.9%.
But
To view the link you have to Register or Login to do it for you. Note that after you have more than 85 people in a group (only 85!), the probability on the calculator reads 100% because the odds are now greater than 99.99%