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### ::: HIDE ::: Birthday Probability, Class of 25

```Date: 11/14/2002 at 16:55:21
From: C.J. John
Subject: Birthday probability

A class consists of 25 students. Find the probability that no two
students will have the same birthday, assuming that all days in a year
are equally likely birthdays and ignoring leap years.
```

```
Date: 11/15/2002 at 21:24:30
From: Doctor Fwg
Subject: Re: Birthday probability

Dear C.J.

This problem, in one form or another, seems to come up frequently, and
the answer almost always seems astounding when first encountered.
There are several other ways to solve this problem but the one I am
listing here is the easiest to understand, I think. You
can also find other related solutions in our archives - search using
the keyword birthday - and in the Dr. Math FAQ:

The Birthday Problem
http://mathforum.org/dr.math/faq/faq.birthdayprob.html

In any case, each person can compare his/her birthday with every other
person, but only once. For example, if there are six people in a room,
the first can compare his/her birthday with five others, the second
can compare with four others, etc. So, for six people, there are 15
possible combinations like this. For n people there are n(n - 1)/2
combinations.

Now, assuming 364 days in a year (ignoring leap years), the
approximate probability of a birthday match between any two people is
(1/364), but the probability of no match is [1.0 - (1/364)], or
(363/364). So, the probability of two "no matches" in a row is:

(363/364)(363/364)  =  (363/364)^2  =  0.9945.

Now, if there are n combinations of possible matches, the probability
of n "no matches" in a row is (363/364)^n.

For 25 people, n = 300. So, for this case, the probability of no
birthday matches for these 25 people (assumming no twins or triplets,
etc.) is (363/364)^300  =  0.4381.

You can see that in this case (of 25 people) the chances are better
than even that at least two of these people will have the same
birthday. So, if you were to survey, say, thousands of different
rooms, each holding 25 randomly selected people (with no twins,
triplets, etc.), you would find that about 56% of these rooms would
have at least two people with the same birthday.

I hope this has been helpful.

With Best Regards,

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

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