How many people do you need in a group to ensure at least a
50 percent probability that 2 people in the group
share a birthday?
Let's take a show of hands. How many people think
30 people is enough? 60? 90? 180? 360?
Surprisingly, the answer is only 23 people to have
at least a 50 percent chance of a match.
This goes up to 70 percent for 30 people, 90 percent for
41 people, 95 percent for 47 people. With 57 people
there is better than a 99 percent chance of a birthday
match!
Presentation Suggestions:
If you have a large class, it is fun to try to take a poll
of birthdays: have people call out their birthdays.
But of course, whether or not you have a
match proves nothing...
The Math Behind the Fact:
Most people find this result surprising because they
are tempted to calculate the probability of a birthday
match with one particular person.
But the calculation should be done over all pairs of people. Here is a trick
that makes the calculation easier.
To calculate the probability of a match, calculate
the probability of no match and subtract from 1. But the
probability of no match among n people is just
(365/365)(364/365)(363/365)(362/365)...((366n)/365),
where the kth term in the product arises from considering
the probability that the kth person in the group doesn't
have a birthday match with the (k1) people before her.
If you want to do this calculation quickly, you can
use an approximation:
note that for i much smaller than 365,
the term (1i/365) can be approximated by EXP(i/365).
Hence, for n much smaller than 365,
the probability of no match is close to
EXP(  SUM_{i=1 to (n1)} i/365) = EXP(  n(n1)/(2*365)).
When n=23, this evaluates to 0.499998 for the probability of no match. The probability of at least one match is thus 1 minus this quantity.
For still more fun, if you know some probability: to find the probability that in a given set of n people there are exactly M matches, you can use a Poisson approximation.
The Poisson distribution is usually used to model a random variable
that counts a number of "rare events", each independent and identically distributed and with average frequency lambda.
Here, the probability of a match in a given pair is 1/365. The matches can be considered to be approximately independent. The frequency lambda is the product of the number of pairs times the probability of a match in a pair:
(n choose 2)/365. Then the approximate probability that there are exactly M matches is:
(lambda)^{M} * EXP(lambda) / M!
which gives the same formula as above when M=0 and n=365.
How to Cite this Page:
Su, Francis E., et al. "Birthday Problem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
