James K. Polk and Warren G. Harding: Two Presidents… both born on November 2nd.
There are 365 possible days that a person could be born, so out of just 44 Presidents,
it’s an amazing coincidence that two would share the same birthday… right?
Actually, no. In a group of 44 people, there’s a 93% chance that two of them could share
a cake. Here’s why.
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First, let’s simplify the problem a bit. Let’s figure out how many people you’d
have to get together in a room to do just better than a 50/50 chance of a birthday match.
…Assuming none of them are twins. Or triplets. Or that you’re not at a convention for people
whose birthday is on like April 7th or something.
You’re the first person in the room. There’s a 100% chance that your birthday is your birthday.
What are the chances that the second person to walk in the room doesn’t share your birthday?
Well, they have 365 birthdays to choose from, and 364 won’t match yours, so, a measly
0.3% chance of a birthday in common. When the third person walks in, there are
363 possible birthdays left that don’t match any birthdays already in the room, but how
do we put these together?
When we combine the odds of independent choices together, we multiply their probabilities.
That let's...
Whoa whoa whoa whoa don’t run away from the math, it’s not that scary. Look! A kitten!
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So the chance that you, person number 2 (364/365) and person number 3 (363/365) having a unique
combination of birthdays, where no one sharing, is about 99.2%
With every new person we add to the room, there’s one fewer birthday available, and
we continue to multiply the combinations.
23 people. That’s all the people we need before you have a greater than 50% chance
of two sharing a birthday.
This is a birthday paradox. It goes against our intuitions… because our brains are bad
at figuring the power of chance. Sure, in that room there’s only 22 possible combinations
of your birthday with someone else’s, but there’s 253 combinations of everyone’s
birthdays.
Our brains have trouble imagining these combinations, estimating things that grow exponentially.
For example, how many times do you think you’d have to fold a piece of paper in half before
its stacked height reached the moon? Well, we fold it once, two sheets, fold it
again, and we have four where we once had one. Now, if I could keep folding this piece of
paper indefinitely by 41 folds we'd reach over halfway to moon, we only need ONE
more fold to cover allll that remaining distance. 42 folds. 42. That number does come seem to
come up a lot.
Ok, so now you understand how few people it takes to get one common birthday, but let’s
make it personal.
What size group would we need to get, say, a 90% chance of one of them sharing your specific
birthday?
Now to... Whooooa hold on! The kitten's back, and this time it's in a box.
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In this case, every new person that we add to the room has the SAME chance of NOT sharing
your birthday. The chances of not sharing combine, and combine
with every new person we add to the room. The chances of them having your exact birthday?
Ah, you’re catching on. Now we want to figure out what that group
size is, that number N. So we can do some rearrangement, take the log of both sides…
To have a 90% chance of one person having YOUR birthday, you’d need 840 people…
I’ve been noticing on Facebook lately that I seem to have at least one friend with a
birthday every single day. But on closer examination, it’s NOT actually every single day.
I’ve got most birthdays filled, but a few are still open.
So how many would I have to have to be able to type “happy birthday” on a friend’s
wall every single day? Hypothetically, you’d only need just 365:
one for every day, but as we’ve seen, the chances of that happening are pretty small.
Now, the first person you add will have a unique birthday. The second person probably
checks another birthday off the list, though there’s a tiny chance they share the first
person’s birthday…. The third person could be a new birthday, or be the same as birthday
1 OR 2. As we go on and on, adding new friends, we
will have fewer open birthday slots to fill, and many of our birthdays will start to have
two, three, maybe five people per day. We check off a large number of birthdays very
quickly, but the last few will take way longer than we’d expect.
If everyone on Facebook added friends until they filled every birthday, the average number
needed would be 2,153. Of course, any individual person may need way more than that. To have
a 90% chance of hitting every birthday, you’d have to add more than 21,000 friends.
So, get to clickin'.