**Ask Marilyn ®** by Marilyn vos Savant is a column in *Parade Magazine*,
published by PARADE, 711 Third Avenue, New York, NY 10017, USA.
According to Parade, Marilyn vos Savant is listed in the
"Guinness Book of World Records Hall of Fame" for "Highest IQ."

In her *Parade Magazine* column of August 3, 1997, a reader asks Marilyn to explain why, in a randomly chosen
group of 50 people, it is virtually certain that at least two will share the same birthday.
Rather than explain why this is true, Marilyn asserts that this is "just plain wrong."
In a followup column published November 23, 1997, Marily admits that in a randomly chosen group of 58 people,
the chances for this being true are 99%.

Charlie Kluepfel <ChasKlu@aol.com> and Hugh Hoskins <hthoskins@earthlink.net> both wrote to point out that in any randomly chosen group of 50 people, the probability is slightly greater than 97% that two will have birthdays on the same date. Since the probability is above 97%, Marilyn was wrong to assume that this well established fact is an "erroneous extrapolation."

For the mathematically inclined, the probability of two people **not**
sharing the same birthday is 365/365 * 364/365.
In other words, there are 364 out of 365 chances that two randomly
selected people will **not** share the same birthday.
(For the sake of simplicity, these numbers neglect people born on February 29.)
For three people, the probability is 365/365 * 364/365 * 363/365.
For four peple, the probability is 365/365 * 364/365 * 363/365 * 362/365.
For fifty peole, the probability is 365/365 * 364/365 * 363/365 * 362/365 * ... * 316/365.
Working out the math, the probability is slightly less than 3% that no two of these
fifty people will share the same birthday, or slightly more than
97% that at least two of these fifty people will share the same birthday.

%bc scale=50 define p(x,y) { auto i,j; j=1; for(i=y-1;i>(y-x);i--) { j=j*i/y; } return(1-j); } p(1,365)0p(25,365).56869970396946388561788409084722390123865271939778p(50,365).97037357957798839991865520436840386584171845099543p(365,365)1.00000000000000000000000000000000000000000000000000quit%

Here are the results for 57 and 58 people, as computed by David Aldrich's bc script:

57: .99012245934116997884528144909275185522219994236295

58: .99166497938926124242286763375497964769434954040491

57: .98998979806519874607615199917004994769423765735635

58: .99154876394029074463806275339766511977464326809594

Perhaps this is where the number 58 came from.

Being that Marilyn doesn't see fit to publish David Pleacher's program for calculating the probability of a birthday match, we can only surmise that Adam Frank Nevraumont's hypothesis is correct, and that 366 days per year were used, with equal likelihood of any of these days. If so, it represents an error. The following program considers any of the 1461 days in a four-year cycle to be equally likely:

DEFDBL A-Z DO INPUT "# of people:", n tProb = 0 ' prob of No Match is first calculated ' calc for # of Feb 29's as 0 or 1: FOR nFeb29 = 0 TO 1 ' above this "No Match" is impossible as 2 Feb 29s would match pOver = (1460 / 1461) ^ (n - nFeb29) IF nFeb29 = 1 THEN pOver = n * pOver / 1461 FOR i = 2 TO n - nFeb29 pOver = pOver * (365 - i + 1) / 365 NEXT tProb = tProb + pOver NEXT p = 1 - tProb ' reverse to make prob of at least 1 match PRINT USING "###.####"; p * 100 LOOP This results in # of people:56 98.8264 # of people:57 99.0062 # of people:58 99.1612

showing that with Feb. 29 birthdays' being 1/4 as likely as any other, there is still over 99% probability that there will be a match among 57 people (as there is about 96% probability that no one will have a Feb 29 birthdate in the group).

A truly rigorous calculation would take into consideration seasonal variations in birthrates. How much the result would be affected is conjectural, but it could only cause the probability of a match to go up, not down.

http://www.wiskit.com/marilyn/birthdays.html last updated June 30, 1998 by herbw@wiskit.com