How does the birthday paradox work?

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📓 The short answer

The birthday paradox shows that in a group of 23 people, there’s over a 50% chance of a shared birthday. Only 23 people seems counterintuitively low, but with 253 possible pairings, a match becomes far more likely than it seems at first glance.

📚 The long answer

How many people must be in a group for there to be a greater than 50% chance that two share a birthday?

The answer? Just 23 people.

Fill the room with 75 people and the likelihood jumps to 99.9% of at least two people sharing a birthday cake.

The birthday paradox is a famous probability paradox because it shows that only 23 people are needed for a greater than 50% chance of a shared birthday—much lower than most people expect. So it's probably easier to understand the paradox if we start with a simpler coin-flipping example.

🪙 Starting simple: Flipping a coin

What are the chances of flipping a coin and landing on heads 10 times in a row?

Each coin flip has a 1/2 chance of landing on heads. Since each flip is independent, you multiply the probability of heads (1/2) by itself 10 times. This gives you:

Therefore, the chances of flipping heads 10 times in a row is about 0.1%.

Now let's flip things around: If the chance of 10 heads is 0.1%, the probability of landing on tails at least once in those 10 flips is 99.9%. Keep this in mind because the same principle applies to the birthday paradox.

🎂 The birthday paradox: Find the probability of no matches at all

Returning to the birthday paradox, it states that there's an over 50% chance of a group of 23 people having at least two people sharing the same birthday.

First we need to find the probability of no one in that group of 23 people sharing a birthday. The formula for that is:

To complete the formula and calculate the chances of there being no birthday matches, let's determine these two missing variables.

What's the probability of it not being someone's birthday?

To keep things simple, we'll assume:

• There are no leap day birthdays.
• All 365 days are equally likely for a birthday.

That leads to a 364/365 chance of a given day not being someone's birthday. So now we've filled in part of the formula:

How many total pairs are there in the group?

If there's a group of 23 people, how many possible pairings can you make between everyone in the group? You could count this all up manually, but let's make our life easier with some math.

First, take the total number of options you have to make a pair in this group. Since you can choose anyone in the group, you have 23 options. Next, determine the total number of options you have for the second person in this pairing. Since you can't select the same person again, you now have 22 options.

So to find the total number of pairings, you take (23 x 22) and divide by 2. The division is important so you don't double count the pairings. For our purposes, Doug paired with Patti is the same as Patti paired with Doug. (👋 Hi, Mom and Dad!)

This shows us that there are 253 possible pairings in a group of 23 people. Let's bring it all together now and fill out these missing variables we had in the beginning:

This formula can now be rewritten (and calculated to be):

But remember that this formula is calculating the chances of no one in that group sharing a birthday. Subtract 0.4995 from 1 to find the probability of at least one birthday match. That results in a 50.05% chance of a birthday match being found in a group of 23 people.

The reason why it seems like 23 feels like a shockingly small group to find a birthday match is because it's hard to comprehend how many possible pairings there are in a group (253). Combination calculations to find pairings are simply tough to intuit, so thank goodness we have calculators.

--- Thanks for reading this week's newsletter! If you have any thoughts, questions, or favorite GIFs, my inbox is always open. Just hit reply to send me a note! :) ​ All my best, Caitlin ​Click here to read five facts about Caitlin​

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​Sources for this week's newsletter​

📖 Book of the week

​Around the World in 80 Words by Paul Anthony Jones​

I happened upon this book as I was perusing my local library, and I'm glad I picked it up. It's a delightful book that whisks you around the world to teach you about the origin stories of 80+ words that have made their way into the English language. If you know me well, you know that I love exploring etymology and knowing the backstories of words. It's a great little book for all my word nerds out there.

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​Check out the full list of books I've recommended here.

👀 Catch up on other curious questions

• ​Why is dental insurance separate from medical insurance?​​
• ​Why are you more buoyant in salt water?​​
• ​Why do birds chirp in the morning?​​

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