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### Johnny Carson's Birthday Problem

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Date: 04/03/2002 at 20:51:48
Subject: Birthday problem

Johnny Carson had a guest on the Tonight Show who told him that the
probability of two people in a group of 50 or more having their
birthday on the same day was very high. Mr. Carson experimented with
this on his show, choosing a guest at random to give his birthday,
after which he asked that any other guest with the same birthday let
it be known. the response was negative, and so he tried again with no
luck then either. I was curious as to why this did not work.

Thank you,
```

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Date: 04/03/2002 at 21:22:59
From: Doctor Ian
Subject: Re: Birthday problem

Here is the reasoning behind the probability.

Assume there are two people. The probability that the second person
does NOT have the same birthday as the first is 364/365, or 0.997.
(Do you see why?)

Add a third person. The probability that the third person does NOT
have the same birthday as either of the others is 363/365. So the
probability that none of the people have the same birthday is
(364/365) * (363/365), or 0.991.

If you keep adding people, when you get to the 22nd person, the
probability that NONE of them have the same birthday is slightly less
than 0.5. That means that the probability that AT LEAST TWO of them
share a birthday is slightly greater than 0.5.

But that's NOT what Carson was testing. He picked a couple of people
at random, and no one else shared THEIR birthdays.  This isn't to say
that there weren't two or more people who shared a birthday that he
didn't pick.

What he should have done, to test the calculation, was to have
everyone in the audience write his birthday on a slip of paper and
pass the slips forward. Then he could have looked through them to see
if there were two slips with the same date. In a room with 50 people,
the chances of finding two such slips are quite good (although
certainly not 100%).

It would be very unlikely, but suppose that there were 50 people in
the audience: one born on January 1, another on January 2, and
everyone else on January 3. If Carson picks the first two guys, gets
their birthdays, and asks for matches, he's not going to find any. But
does that mean that there aren't any matches out there? Obviously not!

Probability can be a tricky subject, and very slight changes in the
description of a situation ('there are two people out there who have
the same birthday' versus 'there is another person out there with the
same birthday as anyone I pick') can cause very large changes in the
associated probabilities.

Also, note that probabilities only tell you what is _likely_ to
happen, not what _will_ happen. It's easy to forget that sometimes.

Does this help?

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

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