Johnny Carson's Birthday Problem
Date: 04/03/2002 at 20:51:48 From: Murad 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, Murad
Date: 04/03/2002 at 21:22:59 From: Doctor Ian Subject: Re: Birthday problem Hi Murad, 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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