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
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%
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