The Birthday Paradox

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Did you know that if you were in a room of 23 people, there is a 50% chance that at least 2 people have the same birthday?

More astonishingly, in a room of 59 people, there is over 99% chance that at least 2 people will have the same birthday.

This is the birthday paradox. It is a probability problem that's actually pretty easy to solve. 


Take a room of 23 people, match 2 people up. The number of ways that 2 people can be selected is:

 23 options for the first person x 22 options for the second person = 506 different matches. 


However, if say Sarah was the first pick and Joe was the second pick, this still ends up the same selection (Sarah and Joe) than if it were the other way round. We need to divide the total number of matches by 2 to get the actual number of combinations. 

506/2 = 253 total combinations

Here is the advanced way using combinatory maths to calculate this. You can skip this part if you like.

Choosing just 2 out of 23 is 23!/(2! 21!) = 23 x 22 / 2 = 253 

Or using nCr notation, 23C2 = 253


Calculate the chance that a pairing have the same birthday:

Sarah could have any birthday in the year. That is, 365 options out of 365.

Joe is restricted to only 1 day in the year (Sarah's birthday), 1 out of 365.

Now, for just this pairing, the likelihood of them having the same birthday is:

365/365 x 1/365 = 1/365

Which means the likelihood of them not having the same birthday is 364/365. 


An Exponential View

The confusion usually starts at this point whether you realise it or not. Most people don't realise that those other 252 pairings make a big difference. However, it might not be in the way in which comes naturally. Think of this statement:

Probability of no one having matching birthdays + probability of at least 1 pairing having matching birthdays = 1

That makes sense right? There are no other possibilities other than these. The first part can only happen in one way, there is not more than one way for "no one having matching birthdays". They either all have different birthdays, or there is a match.

BUT, the "at least 1 pairing" part may have caused some questions, and rightly so. This could mean only 1 matching pairing, or it could mean all 23 people have the same birthday, or 2 sets of matching trio's (2 sets of 3 people with matching birthdays), or any combination between. Baffling right? It gets worse...

Also, 1 matching pairing could have been from any number of matches, not just Sarah and Joe. It gets complex, and we don't like complex. We don't want to work out the individual probabilities, because there are too many.


Think of it this way instead. If the likelihood of any pairing not having the same birthday is 364/365, this is the same for all 253 pairing possibilities. Multiply 364/365 by itself 253 times. In other words:


(364/365)253= 49.95%

So there is a 49.95% chance that all 253 pairings will have different birthdays. Meaning.....there is a 50.05% chance that at least one pairing will be a birthday match. Viola!


The same is true of 59 people (where there are 59 x 58 /2 pairings = 1711 pairings)

Chance of no matching birthdays is:

(364/365)1711= 0.91%


Making a 99.1% chance of their being at least one match. 

Will this make you change the way you view a crowded room? Probably not, but it was worth a go!