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insider
10-22-2002, 02:21 PM
I had heard a long time ago that if you had between 20 and 30 people together your odds of finding two with the same birthday, "month and day" were 85% in your favor. Also, I heard that the same goes for the total amount of pocket change for the same number of people. Is this true or just a math teachers imagination?

Note: I tried to post this same question but the only thing that posted was the word "What". This board takes the longest to post and access than any other I have been on before. I understand that it is busy but sometimes it takes 3 or 4 minutes for it to go from one page to another or to open a post. I think that the gerbils did eat the rest of my post.

Cabbage
10-22-2002, 03:00 PM
It's true. Well, maybe not 85%, but pretty high, and much higher than most would think.

It's easier to find the probability that nobody shares a birthday:

Put the people in some arbitrary order. The first person can have any of the 365 days as his birthday (we're ignoring leap years).

The second person can have any of the remaining 364 out of 365.

The third can have any of the remaining 363 out of 365...

And so on.

The probability that, given n people, no two of them share a birthday is then:

365*364*363*...*(365-n+1)
----------------------------------------------------------
365n

=

365! / [(365-n)!*365n

Subtract this from one to get the probability that at least two share a birthday.

For 23 people, there's about a 50.7% chance that at least two will share a birthday. For 30, it's about 70.6%.

All of this assumes a uniform distribution of birthdays throughout the year. If the birthdays are not distribiuted uniformly, the probability will actually be greater.

A similar thing would be true for the amount of change in one's pocket. I'd imagine most people carry at most \$3 or \$4 in change, so we can assume most would have between \$0 and \$4 in their pocket (401 different possibilities), so the odds would be similar.

Achernar
10-22-2002, 05:43 PM
Hmmm. I've got.... \$7.10, plus six subway tokens. I think I'm unusual, though. I was born on 31 Sep, too. The distribution of change in pockets is probably not uniform, at any rate.

One thing that's important to keep in mind is that this is a very different problem from "What are the odds that one of these 30 people shares my birthday?"