Check your intuition: The birthday problem – David Knuffke

Imagine a group of people. How big do you think the group
would have to be before there’s more than a 50% chance
that two people in the group have the same birthday? Assume for the sake of argument
that there are no twins, that every birthday is equally likely, and ignore leap years. Take a moment to think about it. The answer may seem surprisingly low. In a group of 23 people, there’s a 50.73% chance that
two people will share the same birthday. But with 365 days in a year, how’s it possible that you need such
a small group to get even odds of a shared birthday? Why is our intuition so wrong? To figure out the answer, let’s look at one way a mathematician might calculate
the odds of a birthday match. We can use a field of mathematics
known as combinatorics, which deals with the likelihoods
of different combinations. The first step is to flip the problem. Trying to calculate the odds
of a match directly is challenging because there are many ways you
could get a birthday match in a group. Instead, it’s easier to calculate the odds
that everyone’s birthday is different. How does that help? Either there’s a birthday match
in the group, or there isn’t, so the odds of a match
and the odds of no match must add up to 100%. That means we can find
the probability of a match by subtracting the probability
of no match from 100. To calculate the odds of no match,
start small. Calculate the odds that just one pair
of people have different birthdays. One day of the year will be
Person A’s birthday, which leaves only 364 possible birthdays
for Person B. The probability of different birthdays
for A and B, or any pair of people, is 364 out of 365, about 0.997, or 99.7%, pretty high. Bring in Person C. The probability that she has
a unique birthday in this small group is 363 out of 365 because there are two birthdates
already accounted for by A and B. D’s odds will be 362 out of 365,
and so on, all the way down to W’s odds
of 343 out of 365. Multiply all of those terms together, and you’ll get the probability
that no one shares a birthday. This works out to 0.4927, so there’s a 49.27% chance that no one in
the group of 23 people shares a birthday. When we subtract that from 100,
we get a 50.73% chance of at least one birthday match, better than even odds. The key to such a high probability
of a match in a relatively small group is the surprisingly large number
of possible pairs. As a group grows, the number of possible
combinations gets bigger much faster. A group of five people
has ten possible pairs. Each of the five people can be paired
with any of the other four. Half of those combinations are redundant because pairing Person A with Person B
is the same as pairing B with A, so we divide by two. By the same reasoning, a group of ten people has 45 pairs, and a group of 23 has 253. The number of pairs grows quadratically, meaning it’s proportional to the square
of the number of people in the group. Unfortunately, our brains
are notoriously bad at intuitively grasping
non-linear functions. So it seems improbable at first that 23
people could produce 253 possible pairs. Once our brains accept that,
the birthday problem makes more sense. Every one of those 253 pairs is a chance
for a birthday match. For the same reason,
in a group of 70 people, there are 2,415 possible pairs, and the probability that two people
have the same birthday is more than 99.9%. The birthday problem is just one example
where math can show that things that seem impossible, like the same person winning
the lottery twice, actually aren’t unlikely at all. Sometimes coincidences aren’t
as coincidental as they seem.

About the author


  1. It's easy to comprehend the number of pairs; that's not a problem for me. But I still can't grasp that you need so little people. In a group of 23 people, you'll only have 23 birthdays, so how can they possibly match up? My difficulty is trying to connect the logic of the number of pairs to the raw number of birthdays there are in the group. Could someone help me out?

  2. My statistics professor actually ran this situation on our first day of class. As it happens, my birthday was the same as the person sitting right next to me, who I had never met before.

  3. I remember reading about this in Adam Fawer's book Improbable and I couldn't understand the problem back then as well.

  4. My group of friends have 23 people (including me) 2 people who share a birthday and 2 other people who also share a birthday

  5. Ive been a pharmacy tech for almost a year and ask 100+ people for their birthday each shift and have never helped someone with my birthday.

  6. I have the right intuition I guess. Intuition comes from knowledge, and I have already practiced permutations and combinations and probability

  7. I was 18 years in school had 7 classes of 25 students I knew everyone birthday's and I never had two people having the same birthday. I have two daughters that are in class now around 25 students and none of them have two people having the same birthday. This 50 % thing doesn't add up… Sorry

  8. Can anybody calculate minimum people required so that 2 people have 50% chance of dying same year,Given that most of us won't live more than 100 years. Just for fun 😂

  9. I think the bigger mental hurdle for people to overcome is that they're thinking of the odds that someone in the group shares a birthday with THEM.

  10. The first term 365/365 confuses me. That means the probability of NOT having the same birthday with yourself is 100%?? Seems like it would be 0% because it IS the same birthday. Of course the whole thing would equal 0 with this, so I know I'm missing something here.?.?? Of course multiplying it by 1 doesn't change it which means the term could be removed. The senario with one person isn't needed. I wish I saw the initial setup with 1 and 2 people clearly.

  11. Dear Ted Ed,
    Could you please help me with a simmilar problem?. Actually the session 2018-2019 is going to end soon and I want to know the probality of Shreya(my best friend) and I being in the same class, could you please help me out

  12. Mean while the person who sat next to me today has the same birthday as me
    But the whole school of 11,000 doesn't share it ( the question was asked during a speech as my birthday is a historic day)

  13. The funny thing is. I’m watching this on my birthday and my birthday’s situation right now is the title of this video.

  14. Not only that, but there's also a chance that there are 3 people have the same birthday or even more people. This video hasn't explained that yet but the calculation that substracts 100% by the probability that no people have the same birthday already take this into account.

  15. out of the 2000 or so people in my life that I have ever asked their birthday only 4 have shared mine

  16. Okay so you guys just taught me Permutations and combinations better than any of my math teachers could

  17. Runs towards video: hey a new TED-ed upload! Birthdays, pretty cool, Huh?

    Halfway through video: wtf going over my head smh math SMH

  18. I’ve never met someone who has the same birthday as me. Also, if they’re in the same grade as me, they’re not younger than me. It’s basically a fact now.

  19. I kid you not, for 3 years straight, I happened to share the same birthday as the person sitting next to me. What’s funny is I always find out in the last days of school. (Since my birthday is on June 1st) What’s interesting is that this happened all throughout middle school. The first year was in reading class and on the last day of school. 7th grade was with the clarinetist in front of me. Third year was with my algebra deskie. What a neat surprise it was!

  20. A normal conversation

  21. This really happens in my life. I was at a Japanese restaurant in the local mall , it was full, so they printed out this queue coupon numbered A25. I didnt bother to wait and went to Sizzler one level below, and guess what it was also full and they also printed out a coupon numbered A25 for me wtf!!!

  22. Today is my birthday and nobody celebrated it with me. No family or friends. Then I ended up looking at birthday videos just to spite myself. Hahaha

  23. In my 5th grade class there were not 1, not 2, but 3 people with the exact same birthday!! (November 7th)

  24. how about if 4 people have the same birthday and 2 of those 4 are twins? what are the odds of that? (ps in one of the 2 with no twin. pss it is not my birthday

  25. My brain hurt watching this!! Also I was born on 29th Feb, Leap year baby, have never met another person with same.

  26. I have a class of 48 students
    I have no one with a pair of same birthdays.
    My class is extraordinary.😎😎
    Do you know some other instances?🤔🤔

  27. I’m 1115 and I meet twins every month me my co worker and bar manager share the same bday I know atleast 10 ppl personally with the same bday only thing I can think is Valentine’s Day is 9 months before my bday there’s a few ppl the day before or a few days off .. we are vday babies

  28. 24 people in class and one person is 1 year older than the older. Im born on march 9 while the other person is born on march 8. what are the chances

  29. My birthday was on October 22nd,My mom has a friend who's sons are twins (and my best friends) and their birthdays are also on October 22nd

  30. I understand the maths behind this, but still, in 12 years of school in classes with about 20-25 people there has never been a birthday match. weird.

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