why in a room with 23 people there’s a good chance 2 of them share a birthday
In: Mathematics
You’ve stumbled on something called the Birthday Paradox! The explanation goes a little something like this:
Our goal is to compute the value of P(A), the probability that at least two people in the room have the same birthday.
However, in this case (as with quite a few), it’s actually mathematically easier to compute P(A’) (read as “P of A prime”), the *opposite* proposition — the probability that *no two people* in the room share the same birthday.
This is calculated as follows:
The simple case of one person is simply:
P(A’) = 365/365
that is, of course one person can’t “share a birthday” with people who don’t exist.
If you add a second person, there are 365 birthdays the first person can have, and 36**4** birthdays the second person can have while satisfying our condition.
P(A’) = 365/365 * 364/365
This probability is quite high; it’s very likely two people don’t share a birthday.
You can keep adding people and computing the probabilities as you go; the magic number here is 23. At n = 23, the equation above slips below 0.5, and gives roughly 0.492703.
What does this mean, you might ask? We’re not looking for the odds that no two people share a birthday! We’re trying to find the odds that at least two people do!
Except, we’ve done just that. There are two possibilities: either no two people share a birthday, or at least two people do. The sum of these two cases must add to 1.
Because we now know the odds of one case to be ~0.49, the odds of the other must be ~0.51 — that is,
#In a room of 23 people, there is a greater than fifty percent chance that two of them will share a birthday.
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