What Is The Birthday Paradox?


Oh my god, that’s
my birthday, too. That’s amazing. Not really, though. Hi, I’m Ben. And today’s question
is, what’s going on with the birthday paradox? You’ve probably heard
this one before. The idea that if there
are 20 people in a room, there’s a 50/50 chance
that two of them will have the same birthday. So how can this be? It really is called
the birthday paradox, and it turns out it’s useful
in several different areas, for example, in cartography
and hashing algorithms. You can try it yourself. The next time you’re at a
gathering of 20 to 30 people, ask everyone for their birthday. I mean, don’t be
creepy about it. Play it cool. Say I’m trying to prove the
thing for science, or whatever. And it is likely that
two people in this group will have the same
birthday– not around the same time,
the exact same day. This really surprises people. So the reason it’s
so surprising is because we’re used to comparing
our particular birthdays with some other individual’s
particular birthday. So for example, you
meet somebody randomly and you ask her what
her birthday is. The chance of the two of
you having the same birthday is only one out
of 365, or 4.27%. In other words, the probability
of any two individuals having the same birthday is low. Even if you asked 20
individual people, the probability is still
low, it’s less than 5%. It’s natural that
we feel like it’s very rare to meet
anybody who has the same birthday as our own. When you put 20
people in a room, however, the thing that
changes is the fact that each of these 20 people is
now asking each of the other 19 people about their
birthday simultaneously. Each individual person
only has a small chance, less than a 5%
chance of success. But everyone’s trying
it at the same time, and that increases the
probability dramatically. So the next time you’re with
a group of 20 or 30 people, why not give it a try? You might be surprised. So that’s it. Thanks so much for watching. I hope you enjoyed this. If you are feeling
in a good mood, or you want to learn more
about people’s birthdays, then just go ahead an
click Like down here. Oh, even better, if you have
a suggestion for something you’d like us to answer
in an upcoming episode or some feedback, please leave
it in the comment thread. And what’s the
next– oh, subscribe. I’ll get in trouble if I
don’t ask you to subscribe.

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Comments

  1. I haven't taken a statistics class, but here's my envelope evaluation: 20 people. 1 me. My odds of finding one person are 1/365, which equals .2769.. Every person's odds are .2769.. There are 20 people. Multiply .2769 by 20. 5.5380? that's not very high.. Did I do something wrong?

  2. Out of interest, I put together a script to test this. In fact, I put together two scripts, virtually identical but using different programming languages (VBScript and JScript). Interestingly, JScript gives a little more than 50%, while VBScript gives about 5.4%, but only if I reset the random number generator seed every time I choose a new birthday. If I don't, then VBScript also gives a little over 50%. What does this mean? Is the "paradox" real or not, or is this just bad RNG behaviour?

  3. Have you done anything on 'mystic tears' ? These are the tears associated with emotions. Apparently they are a mystery, not a paradox. That would be good.

  4. so… what's the least number of people that must be in one room such that there is more than a 100% chance that any 2 person share the same birthday?

  5. The intro analogy was wrong, "omg, thats MY birthday too, thats amazing (no its not)" however it is rare, because as you said, the chance of having the same birthday as someone else is 1/365 which is rare. The analogy should of went like, "omg, you have the same birthday as him" which suggests a group of people, which then becomes a likely event.

  6. What I find interesting is that myself and another person can be born at the exact same time, and have different birthdays depending on where you were born.

  7. I have seen the maths explained elsewhere, and I still refuse to believe this concept. It defies all logic; there are 365 potential bday dates!

  8. It would have been cool if you showed the math behind why 20 people asking 20 others simultaneously increase the odds. I can see why, but it would be nice to see the math 🙂

  9. It reaches 50% at 23 people, not 20.
    For those with the patience, here is the math.
    One person enters the room. They have a birthday. The chance of missing the 0 already selected birthdays of everyone else is 366/366, 100%.
    A second person enters the room. There is one birthday in the room they have to miss, with a chance of (366)(365)/(366)(366).
    A third person enters the room. There are now two birthdays in the room to miss, so the combined chance is (366)(365)(364)/(366)(366)(366).
    When the 23rd person walks into the room, they have 22 already selected birthdays they have to miss, with a combined chance of 366*365*364*363*362*361*360*359*358*357*356*355*354*353*352*351*350*349*348*347*346*345*344
    /
    (366)^23.
    Many "paradoxes" and not truly paradoxical. Look up Zeno's paradox, and you will see not a paradox, but some clever jackassery. (Turns out, with quantum physics and the Planck length, Zeno was on to something, but not what he thought, or why he thought.)

  10. so with that logic, 20 people should have around 54% chance to see a couple with same birthday.
    N = 19+18+….+1 is number of different couples.
    (N/365)*100 is our probability.
    remember guys, we are expecting a couple with the same birthday, that couple doesn't mean i am in it.

  11. Been teaching for 4 years, each year with different four classes of at least 25 people. Only encountered twice. The first one two years ago and the second one this year (which involves my birthday :D).

  12. This isn't a paradox, it's just a bit of mathgeek trivia. Hell, most paradoxes aren't even paradoxes. If the Infinite Hotel is full, then EVERY room is taken. Period. Also, there can never be an infinite number of man made objects. Hypothetical situations still need words to mean the same thing as real ones. #triggered

  13. this isn't even true.  when my family gets together it's deffinatly more then 20 people.  and none of us have corresponding birthdate's

  14. The 2 times i did this in college it worked (teachers way of breaking the ice). But there were about 25 people so the odds were a bit higher.

  15. before i watch this (ive heard about 20 people and 50/50 wether 2 have the same birthday)
    well, 20-1=19, 365:19=19.2 (not accurate but this should do), 19+19.2=38,2 and finally 19.2:38,2*100=50,26%
    theres a 50,26% chance of 2 people having the same birthday date. (assuming every year has 365 days, which is of course not the case)

    i think i somehow missed the "paradox" sry :/

  16. This is not a paradox, he himself told at the end there is 0.27% that birthdays are the same. Unless you are japanese who celebrates birthdays on New Year.

  17. I literally just meet a girl with the same birthday as me and In my 22 years of living I've only meet 8 people with the same birthday as– well 8 people that I can remember.

  18. cool video! I'm not a math person but you made this really easy to follow. And, before you comment on how it's not a true paradox, maybe read the description. And, if it's still not true to you, how about some math or a video of your own to back it up? It's easy to say something isn't right and then just move on to the next video but you're not really solving an issue; you're just casting doubt on someone else's credibility and then leaving.

  19. I hate when people say there is a 50/50 chance of that happening. It makes us think 50% out of 100% which sounds a lot better than 35% or20%. But when you stop to think about it doesn't everything have a 50/50 chance of happening? It is either going to happen or it is not right? Lol, I dunno I just think saying 50% applies a greater emphasis on probability then saying 50/50. Stupid right?

  20. It’s not 20, it’s 23 – which actually gives just over 50% probability that ONLY two people share a birthday. Just search youtube for more info. The accurate way is shown on https://www.youtube.com/watch?v=a2ey9a70yY0 . But it’s not surprising as, in a room of 23 people, there are 23×22/2 = 253 different pairs of people to compare birthdays. But 253/365 gives too high a probability because it also includes the probablility of more than two people sharing the same birthday. So watch the video.

  21. This should have been explained with more math examples and visuals…not a paradox but an anomaly I think. good vid.

  22. The distribution of unique birthdays is not random throughout the population. The birth rates are higher starting in July and ending in October due to social and psychological factors affecting the date of conception.

  23. ok.. . . . what is the probability with 20 people, or 30? and this is not a paradox at all.. literally nothing was explained.

  24. my company has 39 people, no same birthday. my hiking group has 27 people, no same birthday. my church choir group has 24 people, no same birthday. Cool paradox.

  25. Here's the actual math behind this:

    The best way to look at this problem is to determine the probability that NO two people have the same birthday. Once we find that probability we can subtract it from one and get the probability that at least two people share a birthday. Here is how we can do it:

    Person 1 has some birthday, Person 2 has a 364/365 probability of not having that same birthday, Person 3 has a 363/365 probability of not having the same birthday as person 1 or 2, Person 4 has a 362/365 probability, and so on down to person 20 who has a 346/365 chance of not having the same birthday as all the others. Now because we're looking for the likelihood of all of these probabilities happening at once we have to multiply all of those chances together which gives us (364*363*362*…*346)/(365^19) which equals about 58.85%. So the likelihood that 2 people in the room do share the same birthday is 100 – 58.85 = 41.15% (not he 50% the video claimed). As a side note, if we wanted it to be a 50/50 shot we would need 23 people.

  26. I think a lot of people are misunderstanding this video. He isn't saying that in a room of 20 people someone is likely to have the same birthday as YOU. He's saying that in a room of 20 people it's likely that any 2 of them will have the same birthday.
    Still not a paradox though. Whoever named this the "birthday paradox" doesn't know what a paradox is.

  27. My bday is Jan 1. My best friend in high school had the same birthday and then I later figured out that his little and only brother also has the same birthday just all different years.

  28. Fucking shit, i was in so much classes in school, and various institutions, and guess what, there no two persons that came with candies in a same day.

  29. My birthday is the 26th of June. I have a friend whose birthday is the 27th, another friend whose birthday is the 28th, and that friend's cousin's birthday is the 25th, and I have another acquaintance whose birthday is the 24th.

    And we all know each other.

    And we were all born in the same year.

    True story! ☺

  30. The chances of 2 people from a group of 20 people having same birthday is (experimentally) 1/2. the chance you are one of those 2 people from the group of 20 is 1/10. multiply these two values to get 1/20, which is approximately 5%. The theoretical probability of you having the same birthday as someone else in a group of 20 ppl is also about 5%. Hence, no paradox.

  31. This video was linked from oddee to explain the birthday paradox, but in my opinion nothing was explained at all…

  32. For all those folks in the comments saying that it isn't a paradox, it's only a "paradox" because our brains can't handle the compounding power of exponents. We expect probabilities to be linear and only consider the scenarios we're involved in (both faulty assumptions, by the way).

  33. Video fails to fully explain and demonstrate why the probability is so exponentially greater. Nor does it explain the paradox

  34. Here’s a different birthday paradox: if you were born for example on June 12, and it was a leap year,366 days would pass between your birth and your first birthday. That means you would have it on June 11? I’m confused

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