why in a room with 23 people there’s a good chance 2 of them share a birthday
In: Mathematics
Birthdays aren’t equally-distributed throughout the year, because most humans get hornier than average during specific seasons. Most children are conceived during these horny seasons, and birthed 9 months later.
While it seems like there should be a 1/365 chance of sharing a birthday with another person, the odds are usually much, much better, particularly if everyone in a room was born in the same area.
Without getting into the math, the way to think of it is number all the people 1-23.
And then think of all the different combos
Person 1 and 12, 6 and 19, 8 and 22… etc at 23 people like the math above shows there is a 50-50 shot of 2 people in the room sharinbf a birthday.
This does not mean that YOU and one of the people share a birthday just that 2 people in the room do.
You’ve stumbled on something called the Birthday Paradox! The explanation goes a little something like this:
Our goal is to compute the value of P(A), the probability that at least two people in the room have the same birthday.
However, in this case (as with quite a few), it’s actually mathematically easier to compute P(A’) (read as “P of A prime”), the *opposite* proposition — the probability that *no two people* in the room share the same birthday.
This is calculated as follows:
The simple case of one person is simply:
P(A’) = 365/365
that is, of course one person can’t “share a birthday” with people who don’t exist.
If you add a second person, there are 365 birthdays the first person can have, and 36**4** birthdays the second person can have while satisfying our condition.
P(A’) = 365/365 * 364/365
This probability is quite high; it’s very likely two people don’t share a birthday.
You can keep adding people and computing the probabilities as you go; the magic number here is 23. At n = 23, the equation above slips below 0.5, and gives roughly 0.492703.
What does this mean, you might ask? We’re not looking for the odds that no two people share a birthday! We’re trying to find the odds that at least two people do!
Except, we’ve done just that. There are two possibilities: either no two people share a birthday, or at least two people do. The sum of these two cases must add to 1.
Because we now know the odds of one case to be ~0.49, the odds of the other must be ~0.51 — that is,
#In a room of 23 people, there is a greater than fifty percent chance that two of them will share a birthday.
Because for any probability we might call “good”, there exists a number X of congregated people that will have a “good” chance of containing at least one pair sharing a birthday.
And since we’re calling 50% our “good” chance, that number is just 23.
Now, what is that probability exactly? It’s a sum of probabilities:
* that all 23 share, plus
* 22 sharing, plus
* 21 sharing, plus
* 21 sharing (and the other 2 form a pair), plus
* 20 sharing, plus
* 20 sharing (and another 2 form a pair), plus
* 20 sharing (and another 3 form a pair)
* …
* only 2 share
Are you tired yet? I am. A lot of times in statistics you might bruteforce a probability by listing every event that would meet the condition. There’s a lot of them. Instead, let’s find the counterpart: the probability that **none** will have the same birthday. This is easier.
The probability that *two* people have different birthdays is 364/365.
If it is given that two people have different birthdays, the probability that a third shares with neither is 363/365.
And the probability that adding up to 23 people, none will share a birthday with the previous, or anyone before that, is of 343, which is (365-(23-1)):
364/365×363/365×362/365×…×343/365
And that’s equal to:
364×363×362×…×343/365^22
Which is [0.4927](https://www.google.com/search?client=firefox-b-1-d&q=364*363*362*361*360*359*358*357*356*355*354*353*352*351*350*349*348*347*346*345*344*343%2F365%5E22).
That is the probability that none will have the same birthday. And to subtract that probability from one tells us the chance that at least two will.
0.50729723432
Or 50.7 percent.
Let’s start with 2 people. Person one has a birthday on one day. So for person 2 to not have the same birthday it is 364/365.
.997% chance to not match. .003 chance to match
Now if we add a third person there is a 363/365 chance of them NOT lining up with the first two. So we multiple 364/365 and 363/365.
.991% chance to not match. .009 chance to match
For the fourth person we do the same. Multiplying the previous chance of there not being any matches by 362/365.
.983% chance to not match. .017 chance to match
As we add more people the fraction we are multiplying by gets smaller and we’ve multiplied it more times so the percentage of not having a match gets high fairly quickly.
Ninja edit:
Here is the formula using factorials ( 3! = 3 * 2 * 1)
(364!/(364-n)!)/365^n
In this n is the number of people. And the result is the chance that they will NOT share the same birthday. At 23 the chance to not share is 46% so the chance they share a birthday is 54%
So while people here are commenting on the probability equation, OP should note that the birthday paradox isn’t meant to be “real”, its meant to simply be a statistics problem and fun equation.
In reality, you actually need LESS people than the equation suggests!
Why? Because the equation describes a world where each day has the same likelihood of a baby being born then. However, in the real world, this isn’t the case, and the days differ in significant manners (and will certainly different in different countries and parts of the world too!)
This gets us to a different solution, that is, since people are more likely to be born on specific days, you are more likely to be able to have LESS people needed in the paradox than the simple equation suggests.
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