365! Is a Big Number

For reasons I won’t get into, I was looking at the Birthday Problem today. The probability of two people out of a group of n people having the same birthday (month/day) is equal to

Birthday Problem Probability Equation

Notice that 365! item? And the subsequent (365-n)! denominator? That pesky “!” means “factorial” and you derive a factorial number by multiplying all of the integers from 1 to n together like this: 1 x 2 x 3 x … x n = n!

So:
1! = 1
2! = 1 x 2 = 2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24
10! = 1 x 2 x … x 10 = 3,628,800
20! = 2.43e18

Obviously these numbers get big fast. I’m familiar with the factorial so when I read “365!” I actually started laughing out loud. I don’t have access to any particular piece of equipment or software that is capable of calculating that number. Thankfully, Wikipedia has a nice table with a short breakdown of the results, depending on n. Now I know that if ten people are in a room, and assuming a normal distribution of birthdates (which is a faulty assumption), the probability of at least two people sharing a birthday is 11.7%. My day is complete.

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7 Responses to 365! Is a Big Number

  1. Bill Ruhsam says:

    If my menory serves me correctly, you need 23 people to have a 50% chance that two will share the same birthday. Just a play on the same theme.

    Dad

  2. James Cronen says:

    When I took Thermodynamics and Statistical Mechanics, we did a lot of calculations with big numbers, and I learned a helpful little trick. You can use Stirling’s approximation to find the factorials of large numbers: ln (n!) ~= n ln nn.

    So, ln (365!) = 365 ln 365 – 365, which is approximately 1788.4. To change to log base 10 rather than ln (log base e), divide 1788.4 by ln(10) to get 776.692. So, 365! ~= 4.92 x 10^776.

    Not a small number.

  3. Annie says:

    And did you know that if you do share your birthday with that person, you are more attracted to them (interpersonally – not just physically) than if you didn’t share a birthday?

  4. James Cronen says:

    @Annie

    That explains why I love Nicole Kidman so much.

  5. Bill says:

    @Annie: What were the conclusions of the researchers about why you’re more attracted to those people?

  6. Annie says:

    @Bill: there are a couple of dynamics in the explanation which boils down (in social psych terminology) to in-group bias and the minimal group.

    First off, people like the people in their in-group better than people in the out-group. They rate their in-group members as more attractive, more intelligent, more [insert positive characteristic] than the out-group.

    Secondly, it takes very little to make a group in a psychological sense. Any shared characteristic will create a certain amount of “group”ness.

    So, as you’ve already illustrated the relative rarity of sharing a birthday with someone else, this makes a pretty distinct group.

    Now this is not to say that this attraction or sense of group are as strong as a group based on more salient personal characteristics. But, it’s been a pretty reliable effect in research.

  7. Chris says:

    Of course, unless you have close to 365 people in the room (and if you did the problem gets a bit trivial), you don’t generally have to calculate anything remotely close to 365! to generate the answer.

    365!/(365-n)! = product of numbers from 365 to 365-n+1

    So for n=10:
    365!/(365-10)! = 365*364*363*…*358*357*356

    This doesn’t require any more mathematical power than the 365^n term.

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