The more people there are, the more opportunities for two of those people to share a birthday. When there are 23 people in a room, the number of opportunities is SO big that the chances of a match between two people is better than half.
So about half of all 23-person rooms should be able to say “Yes, two of the people in this room share a birthday”.
We’re not interested in a specific birthday, such as January 15. We’re asking about the situation where any two people have a birthday in common.
It can help to remember that if there were 367 people in the room, then the chances we have a hit are not just very high, but are actually 100%. EVERY room with 367 or more people has two or more people sharing a birthday. That’s because the only other possibility is that there are 367 unique birthdays in that room, and there aren’t that many birthdays to go around.
What if there were 366 people? Then it’s possible they each have their own birthday (including a Leap Day birthday!), but that’s EXTREMELY unlikely. So the chances of a match are very close to 100%.
What if there were 365 people? Just a bit less. 364 people? Slightly less than that.
… and so on, down to 23. At 23 people, the chances are just over 50%. At 22, they are below 50%. At 2, they are about 1/365, or well below 1%.
The whole thing is just a graph that curves differently than you might expect.
I always explain it with a dartboard example. Put up a dartboard with 365 squares on it, put on a blindfold, and start throwing. You can start to see that randomly hitting it will reduce the space that you can hit that has not been hut before. Humans can “see” that analogy pretty well.
[…] best just to take my word for it. Or try it out here – this page lets you run the a sample set of any size and keeps track of the matches for you. I […]
For those who continue to doubt Mathematical equations (for some reason). I encourage you go to a random number generator website such as random.org and choose the field between 1 and 365. Write the numbers down and see how many you write down before you get a match/repeated number…not long!
Where a lot of people seem to get confused and doubt the equation is that they are stuck on themselves (such as comment 73)…It is not the odds of YOU having the same number as someone else in the room, its the odds of anyone having the same number as anyone, which you explained well. Great site.
[…] intrigue in this paradox (further explained here) is that the chance of having matching birthdays in a group of people seems extremely high for the […]
[…] (The reasoning leading to this conclusion is presented in the paper, and there are many other more detailed explanations of the paradox.) This is a surprising result and highlights how unintuitive probability can […]
[…] The real answer is a measly 23. Once your community grows to 57 people, there is a 99% chance of two residents sharing a birthday. Welcome to the birthday paradox. […]
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I don’t see how the interactive example is correct. If you put 1000 items and only 142 people it says there is a 100% chance of a match. Surely this isn’t correct, for example if person 1 picked 1, person 2 picked 2, 3 picked 3 and so on up to 142 there wouldn’t be a match. Not very likely I know but possible (in fact is it not just as possible as every other selection?) , so there can’t be 100% likelihood of a match. Plus there are many other combination that wouldn’t create a matching pair.
If you keep with the birthday example all you need is 86 people for 100% chance of a matching pair, but it is “possible” for 86 people all to have different birthdays so how can it be 100%.
Sorry if I’m way off the mark, I just don’t get it.
The formula can’t be exact. Using this formula, you would calculate a (though small) chance that in a room of 366 people there would be not a single mutual birthday. This can’t be correct as there are in this case more people than days in a year. Therefore the possibility to have not a single same birthday should be as zero as it could ever be.
Bo, good example. When you have 366 people the probability of everyone having different birthdays has a term which is zero. When you multiply by this zero term you get zero, meaning a probability of one that two people share the same birthday.
When you are calculating, you are working out how many persons DON’T have the same birthday as another. The ‘minus 1’ gives you your chances of another person having the same birthday.
23 individuals x 22 partners = 506 couples. Divided by 2 for unique couples (i.e…Tom and Jane are the same couple of Jane and Tom)
364/365 (Days of the year someone might NOT have the same birthday as you) Yx 253 couples. = 0.4995, being the chances of someone NOT having the same birthday as another person… -1 to view -0.5005…which is only a quick way of seeing the answer when the real equation is 1 - 0.4995 to give you your answer being 0.5005 (50.05% likely of someone having the same birthday as another.
