Given a group of randomly chosen people, how big is the chance that two persons in this group share the same birthday? You can assume that each day of the year is equally probable for a birthday.
The answer might surprise you. In case of a group size of 23 people, the chance that there are two people in there with the same birthday, is already 50/50. A probability of 99% is reached with a group of only 57 people!
The birthday paradox is not a paradox in the sense of a logical contradiction, but is called that because the mathematics differ so much from common intuition. Most people estimate a much lower probability than the numbers listed above.
But don't forget that all we ask is that the birthday of any person in the group matches that of any other person in the group! Not against one particular person picked out in advance. In a group of 23 people, matching the birthday of one particular person with any other person only allows for 22 possible pairs (and thus possible matches). However, when matching any person in the group with any other person, one can make 253 different pairs! Or combinations, if you will. That's what it is called in probability theory and here's how you write it down and calculate it:
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