# What are the odds?

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?

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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:

## 365364363*…*(365-n+1)

365[sup]n[/sup]

=

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

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.

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?”

I don’t have too much to add to cabbage and Achernar’s posts, except a few numbers from an Excel file I created on this very problem once a few years ago (when I was bored, evidently). I throw out February 29 birthdays, too, and assume otherwise uniform distribution.

The probability of a shared birthday exceeds 85% when you have 38 people (86.4%). When you have 57 people, the probability is just over 99%.

Of course, the probability is not 100% until you have 366 people, but it doesn’t take nearly that many in a room to make the likelihood of a shared birthday really damn high.