ELi5: what’s the birthday paradox ? And is its mathematical basics a correct proof?
In: Mathematics
It is the birthday problem since it is not a paradox. The basic problem is to find out how many people is needed for it to be likely that two of them share the same birthday. The chance of two people having the same birthday is one in 365. So you would expect that you need a lot of people to find two that shares birthdays. But it turns out that you only need 23 people. This is because you are not just comparing two peoples birthday, you are comparing all pairs of people among the 23. We do have the full mathematical understanding of this so we can calculate this quite accurately. This problem does appear in several different applications.
The birthday paradox is that, in a room with 23 people, the odds of two people having the same birthday is around 50%.
It is not a *true* paradox, merely a counterintuitive mathematical fact. The proof of it is sound and the issue comes from the fact that when people think of two people sharing a birthday you usually thing of it in terms of sharing a specific day rather than thinking about it being any day of the year.
To start, it isn’t really a _paradox_, but rather a mathematical proof that seems counterintuitive until you understand it.
The birthday problem states that in a random group of 23 people, there is about a 50 percent chance that two people have the same birthday. That _seems_ wrong, because the chance of someone having any given birthday is 1/365.25 – so 23 people shouldn’t be able to get to 50%. You’d need more than that, right?!?
What people miss is that they don’t have to match any one specific birthday, but _any_ of the birthdays of the remaining 22 people. This dramatically chances the probability.
If you want a description of the exact math behind it, [this link is good](https://statisticsbyjim.com/fun/birthday-problem/)
The average person is really, really bad at statistics and probability. The paradox is that people cannot understand simple math, but are still bright enough not to walk into walls.
Human are *terrible* at probability and statistics, which is why the fields have many paradoxes. These are problems that we intuitively think one answer, but careful calculation show a different answer. Human are just bad at this kind of estimation. Here a paradox just mean a result contrary to intuition.
Whether they explicitly think in term of probability or not, most people would intuitively think that the probability for the birthday problem is exactly 1-((365-22)/365) ((365-21)/365) …. ((365-1)/365), which…is a correct answer.
The hard part is actually estimating this number, to see if this is at least 50%. This is the main difficulty, because ((365-22)/365) ((365-21)/365) …. ((365-1)/365) is a product of many factors, each of which are almost equal 1 (for example, the first factor is 343/365 which feels like “basically 1” to most people), so it doesn’t seem plausible that they multiply to a number <0.5. But here we actually have enough of these “almost 1” factor that we multiplied to a number <0.5. Which is why it is a paradox.
It’s not really a paradox. It just goes against human intuition.
It comes from the that if there are 23 people in a room, there is ~50% chance two of them have a birthday in common. Most people would guess the number would be much higher.
It only takes 23 people because the number of pairs of people grows quadratically i.e. if you have *n* people in a room, you can pair them up in n(n-1)/2 different ways, so you have that many chances for a match.
The wikipedia article shows how to calculate the exact probabilities involved: [https://en.wikipedia.org/wiki/Birthday_problem](https://en.wikipedia.org/wiki/Birthday_problem)
The Birthday Paradox asks: “How many people do you need to put in a room before two of them probably have the same birthday?”.
A lot of people immediately say 365, thinking “there are 365 days, so if we put in 365 people, then we’ll probably find two that share a birthday”. The “paradox” is that actual answer is much smaller, though!
The true answer is 23 – with only 23 people in the room, two of them probably share a birthday. The trick is that Person 1 checks their birthday with Person 2, and Person 3, and Person 4, etc…and then Person 2 is *also* checking their birthday with Person 3, and Person 4, etc…and the number of people checking birthdays goes **way** up when you realize that every is checking *every possible combination of birthdays* against everyone else in the room.
It turns out that with 23 people, the odds of two of them sharing a birthday edge up above 50%, which is “probably” when it comes to statistics.
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