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?
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It’s only a *veridical* paradox, which just means that it defies the natural assumptions our brains make.
Briefly, lets look at a coin flip. What are the odds we *don’t* get heads? There is:
0.5 chance to not get heads once.
0.5*0.5=0.25 chance to not get heads twice.
0.5^n to not get heads n times.
We get a basic formula from this for how probability changes for the number of samples. If p is the probability and n is the number, then the formula is p^n
For the birthday paradox, we use the same tactic, but instead ask the odds of two people *not* sharing a birthday.
Because the people are random, each pair has the exact same chance to *not* share a birthday, 364/365.
We can form 253 *unique* pairs of people from a group of 23. If we name them A-W, A has 22 possible pairs, B has 21 since we don’t want to count A again, and this goes on for each person.
So our formula is:
(364/365)^253 = 0.4995 chance they *don’t* share a birthday. This means:
0.5005 chance they *do* share a birthday.
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