# Birthday problem formula is wrong?

## Main Question or Discussion Point

The birthday problem solution on Wikipedia seems to be

P = Prob of having at least 2 people having the same birthday, given n

d = days in a year

n = # of people in room

P = d! / (d^n * (d - n)!)

To make calculations easy, let's assume there are 3 days in a year, thus 3 possible birthdays and 3 people in a room.

According to the formula, the prob of a match should be

= 3! / (3^3 * 0!)

P = 6 / 27, or roughly 22.2% of the time there should be a match.

Does anyone else think this sounds way too low?

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Now, I did it the long way, and I came up with what I think should be the answer

P = 21 / 27 or roughly 77.8 % of the time

I've also run a computer simulation to corroborate my analytical findings, and, in a room of 3 people, there was a match of at least 2 people between 77 - 82 % of the time out of a 100 runs.

Filip Larsen
Gold Member
In your expression for p, p means the probability that no one has the same birthday.

uart
$$\bar{p} = \frac{d!}{d^n \, (d-n)!}$$
The "bar" means it's the complementary probability. So $p = 1 - \bar{p}$ and the correct probability for the question you posed is 1 - 6/27 = 21/27.