I'm Ben Gruber

2013 / 17 May

The Birthday Problem

I regularly read a blog on NYTimes.com called “Me, Myself and Math” which talks about Math in everyday life. Pretty nerdy I know, but you find some interesting tidbits. One that recently came up was the birthday problem. You are always running into birthday coincidences where some one you know has the same birthday as someone else you know. And you think it is incredible. Well, how incredible is it?

Let’s ask a basic question to see. How many people in a group would be necessary to make the odds of two people in the group having the same birthdays 50/50? The answer… 23. What? Much lower than you thought right?

Let’s quickly breakdown why. Let’s say you, your mom and your dad all have different birthdays. That would not be weird, but let’s figure out how unlikely that really was and then we can figure out a bigger group from the same concept.

Okay, to figure your family problem out, we need to think of alternate realities — all the possible combinations of birthdays that could have occurred — and then calculate what fraction of those combinations involve three distinct birthdays.

Okay, starting with you. You have 365 days of open days to avoid having a matching birthday (since you are the only person in the group thus far). Next your mom. She has 364 open days and your dad has 363. If you multiply the number of possible days each person has and then divide that by the total number of days you get the probability of any three people having the same birthday (and also the combination formula*).

So (365 x 364 x 363) / (365 x 365 x 365) = 0.9918 or 99.18%. If you subtract that from one you get 0.82% which are the odds that a group of 3 people would have a matching birthday. To find the answer to the original question, we only need to continue this process.

The equation would look like this:

(365 x 364 x 363 x 362 x 361 x 360 x 359 x 358 x 357 x 356 x 355 x 354 x 353 x 352 x 351 x 350 x 349 x 348 x 347 x 346 x 345 x 344 x 343) / (365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365 x 365)

Don’t believe me? Check out this birthday problem calculator.


*Here is an example of how combinations work taken from the NYTimes Article:

“Suppose you have 3 pairs of pants and 5 shirts. How many different outfits can you create?  Say you decide to wear your ratty blue jeans. Then with five shirts to choose from, that gives you 5 outfits right there. Or you could wear those nice polyester khakis you still have from your high school graduation. Combine them with any of the five shirts and that’s another 5 outfits. Finally, you could go casual and wear your Star Trek sweat pants along with any of the five shirts, creating 5 more outfits and bringing the total to 3 times 5, or 15, outfits in all.

That’s the combination principle in action: If you can make M choices of one thing (like 3 pairs of pants) and N choices of another (like 5 shirts), you can make M x N combinations of them both (15 outfits). The principle also extends to more than two things. If you want to top off your outfit with a stylish hat and you have 6 to choose from, you can create 3 x 5 x 6 = 90 ensembles of pants, shirts and hats.”

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