January 13, 2003
One Hundred Interesting Mathematical Calculations, Number 8:

One Hundred Interesting Mathematical Calculations, Number 8: The Birthday Fact

Look at the person next to you. What's the chance that your birthday and his are different? You would say it is about 364/365--only one time in 365 would your birthday turn out to be the same as hers. Now add a third person. What's the chance that your two birthdays are different, and that his birthday is different from both of yours? Well, the probability that your two birthdays are different is 364/365, and then you have to multiply that by the probability--363/365--that his birthday is different from both of yours.

You can see the pattern developing:

2 people: 364/365 = 0.99726
3 people: 364/365 x 363/365 = 0.99180
4 people: 364/365 x 363/365 x 362/365 = 0.98364
5 people: 364/365 x 363/365 x 362/365 x 361/365 = 0.97286
6 people: 364/365 x 363/365 x 362/365 x 361/365 x 360/365 = 0.95954
7 people: 364/365 x 363/365 x 362/365 x 361/365 x 360/365 x 359/365 = 0.94376
8 people: 364/365 x 363/365 x 362/365 x 361/365 x 360/365 x 359/365 x 358/365 = 0.92566
9 people: 364/365 x 363/365 x 362/365 x 361/365 x 360/365 x 359/365 x 358/365 x 357/365 = 0.90538
10 people: 364/365 x 363/365 x 362/365 x 361/365 x 360/365 x 359/365 x 358/365 x 357/365 x 356/365 = 0.88305

If we continue the pattern by adding more and more people, we find that the probability begins dropping faster and faster. By the time we have reached 23 people, the odds that none of them share a birthday is less than 50-50. By the time we have 41 people, the odds that none of them share birthdays are less than one in ten. And if you have a group of 50 people, you can probably make money by offering odds of 30 to 1 to anybody who wants to bet that there is no shared birthday.

And by 100? By the time we get a group of 100 people together, the odds that none of them share birthdays are down to 0.00003%--only a three-in-ten-million chance of no shared birthdays.

From John Allen Paulos's Innumeracy.

Posted by DeLong at January 13, 2003 05:21 PM | Trackback

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Posted by: leapyear on January 14, 2003 09:16 AM

Posted by: leapyear on January 14, 2003 09:17 AM