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### Probability That Someone Shares Your Birthday

```Date: 05/14/2002 at 16:40:38
From: Sam
Subject: Probabilty Someone Shares MY Birthday?

What is the minimum number of people we would need to assemble in a
group such that the probability that at least one person in the group
has the same birthday as you is greater than 50%?
```

```
Date: 05/14/2002 at 18:26:35
From: Doctor Ian
Subject: Re: Probabilty Some Shares MY Birthday?

Hi Sam,

Suppose you're in a room, and you roll a die, and you get a 2.

Now someone else walks in, and rolls a die of his own.  What is
the probability that he'll get something different than you?
He's got 5 other numbers to choose from, so the probability is
5/6.

If a third person walks in and does the same thing, the
probability that they _both_ get something different from you is
(5/6)(5/6).

If a fourth person does it, the probability is (5/6)(5/6)(5/6),
or (5/6)^3.  Does this make sense?

Now, if the probability that NO ONE ELSE rolls a 2 is less than
50%, then the probability that SOMEONE ELSE rolls a 2 must be
more than 50%, right?

So how many people do we have to cram into the room to get the
probability that NO ONE ELSE rolls a 2 down to 50%?  We're
looking for n such that

(5/6)^n < 1/2

Well,

(5/6)^2 = 25/36

(5/6)^3 = 125/216

(5/6)^4 = 625/1296

which is just a little less than half.  So if you have four other
people in the room, the probability that NO ONE ELSE rolled your
number is less than half... which means that the probability that
SOMEONE ELSE rolled your number is greater than half.

Did you follow that?  You can find out by applying the same
reasoning to your problem.  It's really the same thing, except
instead of 6 sides, each die has 365 sides, one for each day of
the year that might be someone's birthday.  (I'm ignoring leap
years here, to keep things simple.)

It turns out to be a larger than I would have thought!

Can you take it from here?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Probability
High School Probability

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