If you take a group of people born in a non leap year you would need 366 people for a 100% chance that someone shares a birthday but only 23 people for a 50% chance that somebody shares a birthday?
In: Mathematics
For 2 people its 1/366 they share a birthday.
For 3 people. Its 2/366 (1 and 2 share a birthday or 1,3) + 1/365 (or 2 and 3 share a birthday)
For 4 its 3/366 (1,2 or 1,3 or 1,4) + 2/365 (2,3 or 2,4) + 1/364 (3,4)
Etc etc. Do this for 23 people and its around 1/2.
A person can be born in any 365 days. Now if you have another person, they have only 364 possible birthdays so they don’t match any. Third person has only 363 valid birthdays.
By doing this, you could in theory list all the possible alternatives.
Person 1 on Jan 1st, Person 2 on Jan 2nd , Person 3 on Jan 3rd. Hey they are different days!
Okay, how about Person 1 on Jan 1st, Person 2 on Jan 2nd , Person 3 on Jan 1st. Oh, now those two share a birthday.
List down all possible answers. If you have 23 people, and you list all the possible combinations, there are more of the combinations where two people are on the same day, than those combinations where they dont.
Luckily you dont have to list them all you can calculate it. Take the valid combinations and divide by the total.
(365*364*363*…*343) / (365²³)
Okay, so let’s take 23 people in a room and line them up, giving each one of them a number.
Person 1 is then going to compare their birthday to person 2, then person 3, and so on, all the way to person 23. That’s 22 comparisons.
Person 2 is then going to compare their birthday to everyone else in the line except for person 1 (because they already compared, they don’t need to again). That’s 21 more comparisons.
Person 3 will compare to everyone except 1 & 2, for 20 more comparisons. And you keep on going down the line until 22 and 23 compare birthdays.
All in all, you’re going to have 22 + 21 + 20 + 19…..+ 1 comparisons, a total of 253 comparisons.
Each one of those comparisons is going to have a 1/365 chance of having the same birthday. Logically, that also means that each one of those comparisons will have a 364/365 (or about 99.7%) chance of NOT having the same birthday. If you do something with a 99.7% chance of failing enough times in a row, eventually it’s going to succeed.
In this case, we can compute the odds by taking 364/365 and raising it to the power of 253. That comes out to approximately 0.4995, which means that there is about a 50% chance that out of all of those comparisons, none of them will have a matching birthday. And as you add more and more people, that 50% will keep dropping to smaller and smaller chances. But it’s only a 0% chance once you have 366 people, because that would account for every single day of the year, plus one, so there is no possible way for there not to be a match.
The unlikiness of two people being born the same day is countered by the fact that there are a lot of days.
You’re thinking about comparing Person 1 to everyone else and looking for a match, but that’s not it.
You’re comparing Person 1 to People 2 – 23…and then *also* comparing Person 2 to People 3 – 23…and then *also* comparing Person 3 to People 4 – 23…and then *also* comparing Person 4 to People 5 – 23…and then *also…*
It ends up being a much, much, much larger amount of combinations than you thought it was.
Person A and B have a 1/365 chance of sharing a birthday amongst themselves. There’s one possible match.
Add person C to the group, and then A can now match with B as before, but they can also match with C, and B can now also match with C. That’s 3 possible matches
Add D, and then A can match with B, C or D; B can also match with C or D, and C can also match with D. That’s 6 possible matches.
In fact, the number of possible matches increases like this:
(Number of people) x (Number of people -1) / 2.
For 23 people, 23 * 22 = 253 pairs of people who could possibly share a birthday.
With this number being more than half the days in the year, it wouldn’t be more likely to find a pairing that shares a birthday in the group than no pairing shares a birthday.
– Find a 20-sided die (a D20) and start rolling it.
– Every time you roll it, write down the number.
– If you roll a number that you have already written down, stop.
If you roll it twice, it’s pretty unlikely that there’s going to be a “collision”, because you’d need to roll the same number twice in a row.
But what if you’ve rolled the die 10 times already? At that point, it’s a 50-50 shot of rolling a number that you’ve already seen. Much better odds.
First, grammar FYI: it’s not really a paradox, despite the term being used. A paradox is a situation that contradicts itself. There is nothing contradictory about the birthday percentages, its just counterintuitive to many people.
Now to the actual situation. What throws people here is they tend to think only of a specific individual sharing a birthday rather than looking at all the possible pairs.
If you have 5 people in a room there are 10 possible pairings.
* A – B
* A – C
* A – D
* A – E
* B – C
* B – D
* B – E
* C – D
* C – E
* D – E
So even if A doesn’t share a birthday with anyone, the remaining 4 people still might. As the number of people increases the number of pairs increases even more so the possibility that at least two of them match increases more than you would think at first.
The math that goes to show the probabilities for matches gets a bit complicated so its often easier to look at this problem a different way:
What are the chances NO one in the group shares a birthday because there are two possible situations here:
1. No one shares a birthday
2. At least two people share a birthday
Those two events cover every possible situation (including everyone having the same birthday, which is obviously quite rare).
It turns out calculating #1 is super easy.
Lets start with two people.
The probability that 2 people do NOT share a birthday can be calculated as follows:
365/365 (choices for 1st persons birthday) * 364/365 (choices for 2nd persons birthday that is NOT the same as first persons).
The result is 1 * 0.9972 or 99.72% chance that they do NOT share the same birthday. Which makes sense., its a 1/365 chance.
Ok let’s move to 3 people. 365/365 * 364/365 * 363/365 (different than first AND second person).
That’s 1 * 0.9972 * 0.9945 = 0.9918 or 99.18% chance of not sharing a birthday.
Here’s a quick chart:
|PEOPLE|CHANCE NO SHARED BIRTHDAYS|
|:-|:-|
|1|1|
|2|0.9973|
|3|0.9918|
|4|0.9836|
|5|0.9729|
|6|0.9595|
|7|0.9438|
|8|0.9257|
|9|0.9054|
|10|0.8831|
|11|0.8589|
|12|0.833|
|13|0.8056|
|14|0.7769|
|15|0.7471|
|16|0.7164|
|17|0.685|
|18|0.6531|
|19|0.6209|
|20|0.5886|
|21|0.5563|
|22|0.5243|
|23|0.4927|
|24|0.4617|
|25|0.4313|
As you can see the probability of no one sharing a birthday because to decrease significantly the more people you add.
Once you reach 23 people the chance that NO one shares a birthday is only 49.27%, meaning the chance that at least ONE birthday pair exists is 51.83% or greater than 50%
My work department has 23 people. There is one shared birthday. Mine.
I’ve never heard of this paradox but the numbers freaked me out
The best way to think about it is to first realize that when comparing birthdays for 23 people you’re not just making 22 comparisons, you’re making 253.
Why’s that? Because you first compare Person 1 to the other 22 people, that gives you 22 comparisons. You then remove Person 1 and compare Person 2 to the other 21 people remaining, that gives you another 21 comparisons. You then remove Person 2 and compare Person 3 to the 20 people remaining, that gives you 20 more comparisons. You continue this until you’ve compared the birthdays of all 23 people with each other. 22+21+20+19….+3+2+1 = 253
This means that in order for two people to not share a birthday, ALL 253 comparisons need to have no matches. The odds of a single comparison not being a match are 364/365 = 0.99726027 or 99.72%. If you’re making 253 comparisons then the odds of every one of those not matching are (0.99726027)^253 which is 0.4995 or 49.95%. If the odds of no matches between 23 people are 49.95% that means that the odds of at least 1 match are 50.05%.
Ultimately, the reason the birthday paradox doesn’t makes sense at first glance is because people are assuming you’re only making 22 comparisons but when you really lay it out you realize that there are actually 253 total comparisons.
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