woods@hao.UUCP (Greg Woods) (02/13/84)
This is a common fallacy. Just because there are 365 "chances" does *not*
mean the probability of winning is 1. When your "chance" total hit 365 exactly,
that means you have a 50% chance, not a 100% chance, of winning. If you are
going to play that game for money, you better have enough people so that
your "chances" are a lot higher than 365!
This is one of those things where it is easier to calculate the probability
of *losing* and subtract it from one. As you can see, the chance of winning
does not become 100% until you have more than 365 people, which is what your
intuition should tell you. The chance of losing is number of outcomes leading
to a loss divided by total possible outcomes. The total possible outcomes is
easy: if you have N people, it is 365^N (ignoring leap-years for simplicity's
sake. It doesn't change the result significantly). The number of possible ways
to lose is a little more tricky, but not too bad. The first person can have
any of 365 birthdays. The second, if you are to lose, can have any of the
remaining 364. The third, 363, and so on. Therefore the number of ways to lose
is 365*364*....*(365-N+1). This can be expressed mathematically as
365!/(365-N)! where "!" means factorial, fact(n) { n=0 ? return(1) :
return(n*fact(n-1)); }
Therefore, your chance of losing is (365!/(365-N)!)/(365^N). This correctly
predicts your chance of losing to be 1 if N=1, and nearly 0 if N=365.
The formula requires defining -1! if N>365, but surely you can see that it
really is impossible to lose if you have 366 or more people! :-)
GREG
--
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!hao!woodsdarrelj@sdcrdcf.UUCP (Darrel VanBuer) (02/13/84)
The probability of adding the m-th person to a set of people with different
birthday is (roughly) (366-m)/365 [ignores leap year, assumes all dates
equally likely]. So the probabily of a set of m all different is
(364 * 363 * 362 * ... 366-m)/365**(m-1). This works out to about 50% at 25
people and not quite certain at 50 people.
--
Darrel J. Van Buer, PhD
System Development Corp.
2500 Colorado Ave
Santa Monica, CA 90406
(213)820-4111 x5449
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VANBUER@USC-ECL.ARPAsrini@ut-sally.UUCP (Srinivasan Sundararajan) (02/21/84)
The Pigeon-Hole Principle ========================= Why bother with probabilities at all if all you need to be sure is a 100% chance of winning. For those have not heard of pigeon-holing : 1) Assume 365 pigeon-holes - one for each day of the year. 2) Assume P persons in the room. 2) Each person claims a pigeon-hole for himself. ( 2 persons might have the same birthday and thus claim the same hole ). 3) If P = 365, it is possible ( rather slim ) , that they all choose different holes, BUT, if P >= 366, then we are guaranteed that at least 2 persons pick the same box. From: srini@ut-sally