09 September 2006

Things that fascinate me: the birthday paradox

I'm not much of a mathematician but this is a good one for you: How many people need to be in a room such that the probability of two of those people having the same birthday is greater than 50%?

This moderately famous math problem is known as the birthday paradox, even though it isn't technically a paradox. As Wikipedia puts it, "it is described as a paradox because mathematical truth contradicts naive intuition," in other words, the number is much smaller than people believe.

Before you read on, take a guess. How many people need to be in a room such that the probability of two of those people having the same birthday is greater than 50%? The answer is as follows.

Assumptions: we remove February 29th from the problem (sorry leapers); birthdays are spread equally among the remaining 365 days (apparently they are not, but I haven't found data to back this up, yet).

The problem is more easily asked in the opposite direction: what is the probability that no two people will share the same birthday?

364/365 = 0.997, or 0.3% chance that of two people they will share the same birthday.

For three people:

(364/365) x (363/365) = 0.992 (0.8%)

For three people:

(364/365) x (363/365) x (362/365) = 0.983 (1.7%)

And so on. As you can see, as more people are added to the room, the probability that no two of them share the same birthday decreases, and conversely, the probability that two of them will have the same birthday increases in turn.

Warning: answer follows!

It turns out that for 23 people in the room, the result is 0.493, or conversely, a 50.7% chance that two people in a room of 23 people will share the same birthday.

How close were you to the actual answer?
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