Birthday Paradox

[Source]

• If 23 people are in a room together, there is about a 50% chance that two of them share a birthday.
• If 48 people are in a room together, there is about a 95% chance that two of them share a birthday.
• If 88 people are in a room together, there is about a 50% chance that three of them share a birthday.
• In a crowd of 1813 there is a 50% chance that thirteen of them share one birthday.

Why?

When you put 20 people in a room, however, the thing that changes is the fact that each of the 20 people is now asking each of the other 19 people about their birthdays. Each individual person only has a small (less than 5%) chance of success, but each person is trying it 19 times. That increases the probability dramatically. [1]


Here is Khan's more complicated explanation.

 Also:

Several years ago, Diaconis and Fred Mosteller of Harvard University derived a formula that covers multiple matches involving multiple categories. For example, what are the chances that three members of a family have a birthday on the same day of a month (though not necessarily the same month)? Taking the number of days per month to be 30, the formula gives the approximate answer that a triple match in day of the month has about a 50-50 chance if at least 18 people are included in the group.

Now, suppose that certain coincidences involve matches that are close but not exact. It turns out, for example, that it takes just 14 people in a room to have even odds of finding two birthdays that are identical or fall on consecutive days. Among seven people, there is about a 60 percent probability that two will have birthdays within a week of each other. Among four people, the probability that two will have birthdays within 30 days of each other is about 70 percent.

"Changing the conditions for coincidence slightly can change the numbers a lot," Diaconis and Mosteller contend in a 1989 paper in the Journal of the American Statistical Association. "In day-to-day coincidences even without a perfect match, enough aspects often match to surprise us."

What about the fact that birthdays aren't actually uniformly distributed throughout the year? In the United States, the data show a seasonal pattern, varying between 5 percent above and 7 percent below the average daily frequency. [2]