# The Birthday Paradox

Hey, what’s up? It’s Matt and have you ever heard of the birthday paradox It’s the idea that if you have 23 people in a room together the odds that at least two of them share a birthday are Greater than 50% I remember when I first heard this. I was like no way that can’t be right There’s no way with 365 potential birthdays that only 23 people guarantees you a greater than 50% Chances there being a repeat no way even the name of the problem the birthday paradox Reflects the fact that it’s so counterintuitive for so many people so let’s look at this problem together and see how unbelievably, it’s actually true instead of starting with all 23 people and finding the odds that at least two of them share a birthday we’re actually

gonna Take this step by step, person by person. and see the odds that no one shares any birthday For example if person one walks into an empty room the odds that his birthday will be unique meaning it won’t match with anyone elses are 365 out of 365 because no one else is in the room But when the second person walks in the odds that his birthday will be unique are 364 out of 365 because person one’s birthday could match. When a third person walks in the room the odds that his birthday is unique are 363 out of 365. And this pattern continues as more people come into the room Assuming no birthdays have overlapped yet now these here are the odds per Individual that they don’t share a birthday with those already in the room, but we want to find the odds that ALL the individuals; Everyone in the room doesn’t share a birthday. To find that we multiply all the individual odds together This gives us the probability that no one shares a birthday So to find the probability that at least one birthday is shared we subtract this number from one. Now with five people in the room The odds of this happening are pretty low But this number increases really quickly as more people are added: 10 people bring the odds to 12 percent 15 people bring the odds to 25% and 20 people bring the odds of at least one shared birthday to 41% Finally 23 people bring the odds that there are no repeat birthdays to 49.27% Which means the odds that there is at least one repeated birthday are 50.73%. And as you keep adding people to this room the odds that at least one birthday is shared still keep increasing at a fast pace After 23 people gives us 50% it only takes about 20 more to get us into the 90s and by 70 People there is an incredibly small chance that a birthday isn’t shared by at least two people. Now you’ve seen the mathematical information (the proof) that shows you this is real, but it still feels weird right? Even though you’ve seen the math It’s doesn’t feel right that only 23 people gave a greater than 50% chance of their being at least one repeated birthday? well You’re not alone and this actually relates to a criticism of the birthday problem Levied by mathematician Paul hamos. “The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation What calculators do not yield is understanding or mathematical facility or a solid base for more advanced generalized theories” or; Said another way “people are focusing so much on the math of the problem, they don’t understand why it makes sense.” And he’s right. I explained a very mathematical mechanical way of viewing this problem and while it’s accurate that explanation doesn’t help people understand conceptually what’s going on It’s easy to say “let’s multiply some numbers together”, but what does that actually mean? The key to understanding, actually understanding, this paradox Is that there are many more birthday comparisons and therefore potential repeats than we realize. See, when we hear this problem, we typically think of only one set of comparisons. “What’s the likelihood someone else was born on this date?” If there are only 23 people then there are only 22 chances for this to happen. But the key piece to understanding this entire problem is that while there are only 22 chances for 1 specific birthday to repeat We’re looking for the probability that ANYONE’S birthday repeats Not just any specific one and when we understand this it becomes obvious that there are way more than 22 comparisons. When we consider all possible combinations of 23 birthdays, we actually have 253 comparisons. 253 chances to find a repeat birthday. And do you remember what the odds were that the second person walking into our room would have a unique birthday? It was 364 out of 365 Those are the odds of any two people not sharing a birthday Well when we raise it to the power of the number of comparisons we ultimately make, we get the same answer as before. So, there you go. You now know the two reasons why this paradox is Actually Accurate. There’s the unintuitive mathematical way we talked about first, But you should also now understand the conceptual reasoning behind it. This occurs because we are able to make way more xomparisons than there are people in the room, and it doesn’t matter whose birthday is repeated as long as 1 person’s is. Aaaand that’s it! Thanks for watching If you enjoyed this I made a second video on my other channel Everything else. You can click up here-ish to check that out. It’s basically using some of the same methods we talked about in this video, to solve other related types of math problems, and I thought it was cool. So check it out if you want to. Otherwise, that’s it. I’ll see you later. Thanks. Bye! (Subs by NotaNerdfighter)

First

The Monty Hall Problem is going to blow your mind.

There are a lot of related misunderstandings that apply to everyday real life things. Like when the odds of something happening to SOMEONE, anyone at all, are one thing, and the odds of it happening to YOU SPECIFICALLY are another. For a toy example: the odds of someone at all winning a given raffle are 100%: some name or another is going to be drawn from the hat. But the odds of YOU IN PARTICULAR winning a raffle are much lower, decreasing as the number of participants increases.

Survivorship bias tends to factor into this in real life too. People say, "look at all of these people succeeding against apparently-improbably odds all the time! I'll just do that and I'll probably succeed too!", neglecting to compare how many people DIDN'T succeed when trying that, and so what the (low) odds of any PARTICULAR random person (like you) succeeding are, vs the (much higher) odds that just SOMEONE or another (but probably not you) will succeed.

I understood it to the point of "we add the odds together". Cool. Then we substract them from one. Why tho? I'm terrible at math so that part… kinda seems arbitrary

I'll try to explain how I understand it, but mind you that I am still a probability noob as well.

Your second calculation doesn't work out because you failed to account for each additional person in the room shrinking the probability space. IE, following your second method, there would still be a non-zero probability that there isn't a shared birthday at 367 people.

If there are zero people in the room, then any two random people selected from the room will definitely have the same birthday.

Does anyone here share my birthday? 12/31

I knew of the mathematics of this thing before but seeing the diagram made it make more sense why it’s true

whenever two numbers that should be the same arent, i just claim rounding error and ignore it ¯_(ツ)_/¯ (even if it's a bigish difference like .7 or like 5, still totally rounding error)

Hi Matt. I had heard this before and kinda followed the math, but you did a really good job of presenting the info in a way that I can actually understand it. I think.

Now, what do ya know about quantum mechanics? 😃

Here's the paradox flipped on its head: It's

incrediblyunlikely that365 people in a roomall havedifferent birthdaysCalling it a paradox seems weird. Aren't paradoxes usually things that cannot be explained?

On the last boat I served on in the Navy one of my duties was UPC (Urinalysis Program Coordinator). As the UPC I had a database of all 140 crew members and the database included their birthdays. I had learned of the birthday paradox several years earlier so late one night on duty I played around with the database to find all the matching birthdays. There were 22 pairs of people who shared a birthday, 4 groups of 3 people who shared a birthday, and one group of 5 people who shared a birthday.

The method shown at the end (around 4:09) excludes scenarios whereby 3 or more people share the same birthday, which is why the resulting probability is a bit lower. For instance, consider N=3. There are 365^3 = 48,627,125 scenarios for 3 birthdays. The first method described in the video demonstrates that there are 365 P 3 = 365!/(365-3)! = 365!/362! = 365*364*363 = 48,228,180 scenarios whereby 3 people have no shared birthdays. The last method described in the video depicts 364^3 = 48,228,544 ways for the birthdays not to have exactly two matches. The difference between these two numbers (48,238,544 – 48,228,180) = 364 represents situations whereby the three people all share the same birthday. Nonetheless, I do not know why I have 364 and not 365 at the end, if someone could show me my mistake, I'd appreciate it. http://www.wolframalpha.com/input/?i=364%5E(3+combination+2)+-+(365+permutation+3)

But not all 365 days of the year have equal amount of babies some days there are less people born some days a lot of people are born

In my (admittedly short) 16 years, I've met/know of precisely 4 people who share my birthday

Love your content still man! Keep it up. Also love that nerdfighteria flag! DFTBA!

my birthday is shared with the ROCK and david beckham: may 2

Just last week, I found that a person in my Spanish class has the same birthday as me…weird(My bday is 03/03)

This is really cool!! We actually did this same problem on a homework for my AP Calculus AB class, but we only talked about it computationally, not conceptually. This video did an awesome job filling that understanding gap for me. Awesome content! (as always)

To me, this was never really a paradox, since I'm biased in a way that in my elementary school class of about 20 people, there were at least two people with the same birthday (I think it have been even more shared birthdays). Coincidentally, my cousin and two persons I met later also shared that exact birthday [and I don't really know many birthdates…]. Even more interesting: All of these people except my cousin were even born in the same year.

And somehow my family managed to have birthdays close together: My mother's birthday is 3 days before my brother's, my aunt's birthday is the day after her son's which is coincidentally the same as my uncle's (not her husband) birthday…

I'm trying to figure out how to implement this idea into a party ice breaker.

In a room of about 40-50 people I could give a condensed version of this paradox. The first person to find an original pair of bdays wins a prize.

Ideas?

So my birthday is 10/21 and I’ve met 1 person that shares my birthday

Thanks, wonderful explanation!

Hello. The paradox of birthday is very well known but refers to at least two people.

Then, what is the probability if at least 3 people or "N" people different from 2 have the same birthday? Thank you.

To get the right answer you would have to take 1 – "e" to the power of – possible combinations divided by a year.