Re: Saline warm springs and birthdays
Posted by Curious on April 21, 2004 at 12:14:45:
In Reply to: Re: Saline warm springs and birthdays posted by possum That's Right! on April 20, 2004 at 15:01:12:
You are on track as to the value being just over 50%. We have to agree that these are 24 people who are not related (such as twins), we are looking at birthday without the year being improtant and none were born on any particularly odd day, such as that caused by leaps years.
Let:
M = Match, the probability that at least two people have matching birthday.
F = Failure, the probability of no matching birthdays occurring in the group.
How do we calculate M and F?
F will be the number of possible non-match combinations, divided by the number of possible birthday combinations in the population.
The percent probability of a match will be what is left when we subtract the probability of no matching combinations. Easy!
M = 1 ¨C F
Lets find out the probability of no one having the same birthday.
Pretend all the people are standing in a line and you are moving down the line right to left. The first person you encounter has one of 365 birthdays. The second has one of 364 possible birthdays, since one of the 365 days in the year was the first person¡¯s birthday. The third person has one of 363 and as you can see the number of possible birthdays for each individuals is one less then the person before them.
The number of possible non-match combinations in 24 people:
(365)(364)(363) . . . .(365 ¨C(23)) = 1.44327E+61
The total number of possible birthdays combinations (including matches) is simply going to be 365 raised to the power of the number of people, in this case 24. (^ denotes the exponent).
365^24 = 3.12629E+61
Therefore:
F = 1.44327E+61/3.12629E+61
F ¡Ö 0.461655742 or about 46%
There is a 46% chance no one had the same birthday.
As previously presented, the probably of a match is simply calculated:
M = 1 ¨C F
M ¡Ö 1 ¨C 0.461655742
M ¡Ö 0.538344258 or about 54%
The probability of 2 people out of 24 at Saline Hot springs having the same birthday is about 54%!
Woo hoo!