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
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.
Latest Answers