So the birthday paradox is where if you’re in a room with 23 other people there’s a 50% chance of at least two people having the same birthday. Alternatively, In a room of 75 there’s a 99.9% chance of at least two people matching.
Why is this?
In: 134
It seems counterintuitive that only 23 people need to hit a 50% chance. But it’s not about the 23 people, it’s about all the combinations of *pairs of people*. There are (23 x 22) pairs, or 253. That’s a lot of pairs, so there’s a good possibility of at least one of those pairs having the same birthday. And the more people you add, the more pairs you have, and the higher the chance you get.
For simplicity, we’ll pretend leap years aren’t a thing, so there are 365 possible birthdays, with equal probability.
If you have only two people what is the probability that they have the same birthday? It’s 1/365. That means the probability they have different birthdays is 364/365. Let’s say that the second option is true.
A third person enters. What is the probability they have a birthday different from the two already there? Well there are 363 possible birthdays remaining that would fit that criteria, so it’s 363/365.
We can therefore calculate the probability of 3 random people *not* sharing a birthday as *364/365 * 363/365*, which can also be written as *(364 * 363) / 365²*. The answer is a little over 0.99, or 99% (Leaving a less than 1% chance that any two of the three share a birthday).
But we can keep going, adding one more person at a time, with their likelihood of not matching any of the previous going down each time. The result will be *(364 * 363 * 362 … ) / 365^n*, where n is one less than the total number of people (and there are n terms multiplied together).
If you do this for 23 people, it is *(364 * 363 … * 344 * 343) / 365²²*. The answer is just over 0.49, or ~49% chance of no matches. This leaves a ~51% chance of at least one match.
Performing the same calculation for 75 people (n=74) gives you 0.00028. That’s a 0.028% chance of no match, and therefore 99.972% chance of at least one match.
It’s not a true paradox where things seem like they must be both true and false simultaneously. Instead, it uses the term paradox because it is very unintuitive and most people can’t wrap their heads around it.
The birthday paradox happens because people look at 23 people and only consider the odds of the 23rd person sharing a birthday. In actuality, you have to consider *every* pair of people and whether or not they share a birthday.
The 2nd person has a 1/365 chance of sharing a birthday with the first person. Assuming they don’t, then the 3rd person has a 2/365 chance of sharing a birthday with either of the first two. The 4th person similarly has a 3/365 chance of sharing a birthday with any of the first 3 people. If you do all the math (which involves some stuff like flipping it into odds of *not* sharing a birthday and then taking the result away from 100%), you get to a >50% chance at 23 people.
Another way of looking at it is the number of pairs of people. When you have 2 people, you have 1 pair. When you have 3 people, you have 2 pairs. At 4 people, you have 6 pairs, and with 5 people you have 10 pairs. This keeps growing at an alarming rate. At 22 people you have 231 pairs, and at 23 people you have 253 pairs. While the odds of a single pair of people not sharing a birthday is >99%, if you multiply those odds together 253 times you get down to 49% chance. By the time you have 75 people in the room, there are 2775 combinations of people, so the odds drop to nearly 0.
essentially its not a 50% chance that 1 person in particular shares a birthday, its the chance any 2 people have a matching birthday, so you have 22+21+20+19…+2+1 = 246 chances.
The odds are so high because you’re not checking to see if *you* have the same birthday as some else in the room. Instead you’re checking to see if person A has the same birthday as person B, or C, or D, etc…and then *also* seeing if person B has the same birthday as person C, or D, or E, etc…and then *also* seeing if person C has the same birthday as person D, or E, or F, etc…and then *also* seeing if person D has the same birthday as person E, or F, or…
As you see, the number of combinations that you’re checking explodes *much* more quickly than you might first thing. As it turns out, with only 23 people you end up checking *so many combinations* that it’s actually a good bet that at least one of them will be a match.
You are not looking for another person who matches **your** birthday necessarily. You are looking for *any* two people in the group who share a birthday. So the odds of **any pair** matching is the cumulative odds of each person. You start with 22/365 odds of at least one person and you have to add on everyone elses odds (slightly reduced).
With one person there is a 365/365 chance they’ll have a new birthday
At two, there’s a 1/365 chance they’ll have the same birthday, or a 364/365 chance of a new one.
The third has a 2/365 chance of sharing a birthday with a previous person, or a 363/365 chance of having a new one.
Etc
To get the probability of someone sharing a birthday you multiply the odds of each person not matching a birthday together, and then subtract that from 1. So with 5 people you end up with 1 – [365•364•363•362•361 / 365⁵] to get 2.8% odds
The formula for calculating this out would be {1 – [ (365! /(365-n)!) / 365^n ] } where n is the number of people
I actually made a google sheets doc awhile ago to test this. It isn’t exactly accurate as it treats everyday as equally likely to be a birthday, but it was a neat visualization of it. I’m not sure if the link will work, but here goes (refreshing the pages changes the numbers).
[Drive Link](https://docs.google.com/spreadsheets/d/14TAtWgYX3AmjoskiM4PIjDGTc58oWtYxNsTnezbHK1c/edit?usp=sharing)
As others note, the birthday paradox describes a certain counterintuitive behavior of statistics when applied to probabilities that involve pairs. It is important to realize it is *not* intended to be a literally applied phenomenon – but rather more of a thought experiment.
Let’s start with the assumption we have some population of people for whom their dates of birth are independent and uniformly distributed – note this isn’t really true in the real world, but it makes the analysis much easier. Now we can ask various questions about the people in the room like:
>What are the odds a randomly selected person is born on August 17th?
We can answer this question by enumerating all the possible events for our person:
>{ Jan 1st, Jan 2nd, Jan 3rd, … , Dec 29th, Dec 30th, Dec 31st }
From our assumptions, we know that the odds for any particular one is 1/N where N is the size of the set – e.g., 1/365. So far so good, everything makes intuitive sense.
Now let’s ask the more complex question
>What are the odds two randomly selected people are born on August 17th?
We answer by the same process – we’ll enumerate all the birthdays for the first and second person selected:
>{ Jan 1st and Jan 1st, Jan 1st and Jan 2nd, Jan 1st and Jan 3rd, … Dec 31st and Dec 29th, Dec 31st and Dec 30th, Dec 31st and Dec 31st}
Counting up the size of the set we see that it is a total of 365*365 events, of which only 1 satisfies the requirement – both born on Aug 17th.
Now we can ask a slightly different question:
>What are the odds two randomly selected people are born on the same birthday?
The enumerated set of dates is the same as before, except this time our criteria is different – we’re now looking for *any* matching dates, not just a *specific* matching date, e.g.,
>{ Jan 1st and Jan 1st, Jan 2nd and Jan 2nd, Jan 3rd and Jan 3rd, … Dec 29th and Dec 29th, Dec 30th and Dec 30th, Dec 31st and Dec 31st }
There is exactly one match for each day of the year – hence a total of 365 matching birthdays. As above, our enumerated set had 365*365 total dates, so the odds of landing on a matching birthday are 365/(365*365), which reduces to 1/365.
This latest result is counterintuitive – just by changing the selection criteria from a specific date to a matching date, the probability went up dramatically! This is the core truth behind the birthday paradox – our intuition for probabilities in English isn’t very good at capturing the distinctions that matter in the probabilities.
[https://www.reddit.com/r/explainlikeimfive/search/?q=birthday%20paradox](https://www.reddit.com/r/explainlikeimfive/search/?q=birthday%20paradox)
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