The Birthday Paradox Explained

Birthday paradox

Q: How many randomly-chosen people would need to be in the same room to virtually guarantee two of them share the same birthday?

A: 75 (99.9% probability)

This unintuitive answer is known as the birthday paradox.

The earliest known publication of the birthday paradox was in 1939 by the mathematician Richard von Mises. However, there is evidence it was known at least a decade beforehand by Harold Davenport who didn’t claim it as his own because “he could not believe that it had not been stated earlier.”

So why is the answer so unintuitive?

Partly it’s because humans have a tough time with probability and exponentials. Remember the story of the lily pads? But also, because it’s slightly complicated…

[Warning: Math] The chance that two people share the same birthday is low; much less than 1% (1/365 = 0.27%). But there are 75 people involved so you have to check each of them with everyone else. The first person compares their birthday with 74 other people, but the second person only needs to compare with 73 other people (the first person was already checked), and so on. If you add up all possible comparisons (74 + 73 + 72 + … +1), the sum is 2775 comparisons which makes the probability much more likely. I’ll spare you the rest of the math but you can read about it here or watch this excellent video:

You can use the birthday paradox to create an unusual ice-breaker game. Or, you could use your knowledge of these probabilities to win a bet. If there are only 23 people in the room, the odds two of them have the same birthday is 50%. You may not want to bet on that. But if there are 30 people in the room, the odds improve to 70% – that’s a pretty good bet. Here’s a convenient calculator, if you want to try it out.

Math can be unintuitive but understanding it can help you make better decisions. Like which size pizza to order on your birthday. It’s always the bigger one – that’s not much of a paradox.

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5 Responses to The Birthday Paradox Explained

  1. Dr Mike February 19, 2024 at 3:59 pm #

    There’s another version I’ve heard of which involves people picking letters of the alphabet. You only need 13 people to have a 95% chance of a match .

    • Jonathan Becher February 19, 2024 at 4:41 pm #

      This is a good version if you have a small group of people. With only 7 people, you have better than 50% odds of a matching letter.

  2. Brian Etheridge February 19, 2024 at 4:28 pm #

    I like the “circumference v height” bar challenge. On a standard pint (or pretty much any soda glass), challenge the circumference of the top (or often the bottom) is longer than the glass is tall. Visually, that looks impossible. But remember the formula involves multiplying by 3.14. Anything with more than a 1.5 inch diameter is likely to net you a free drink

    • Jonathan Becher February 19, 2024 at 4:38 pm #

      Love this example, Brian. It’s definitely a bar bet you can win. Unless it’s one of those super-tall german steins…

  3. Dustin Mulligan May 13, 2024 at 3:00 pm #

    I think the birthday problem has a much bigger impact when you tell people that only 23 people are needed for there to be more than 50% chance 2 will share a birthday. To me that sounds a lot more bizarre. In fact I’ve conducted this test several times in group chats online, on my YouTube channel. When I had more than 24 viewers on a stream I would get them to start stating their birthdays and half the time we would get a match and everyone would be blown away haha.

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