Birthday Paradox

Discounting leap-year there are 365 days in a year. If you gather 57 people at random, what are the odds that two of them share a birthday? Doesn't seem very likely since there are 7 times more days in the year than the 57 people so lots of space in the calendar to be unique. And yet the answer is surprising....

The odds are 99% that two people in 57 will share a birthday. This is known as the birthday paradox.

For just 23 people the odds are still 50% of a common birthday.

How can this be?

The math works like this...

Let's calculate what must happen for every birthday to be unique.

Person 1 declares his day. It can be any day and he'll be unique so far.  365/365 = 1
Person 2 declares a day different than person one. He/She has the option of 364 days so the odds are 1 * 354/365 = 99.7 % of no match.
Person 3 has the pick of 353 days.
Person 4 has 352 days available
and so on.

With 23 people the odds of all unique birthdays is
365/356 * 364/356 * 363/356 * 362/356 * ... * 343/356 = 49.2%

The opposite of this, 100% - 49.2% = 50.8%, is the odds of a shared birthday amongst 23 people.