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 not a true paradox where things seem like they must be both true and false simultaneously. Instead, it uses the term paradox because it is very unintuitive and most people can’t wrap their heads around it.
The birthday paradox happens because people look at 23 people and only consider the odds of the 23rd person sharing a birthday. In actuality, you have to consider *every* pair of people and whether or not they share a birthday.
The 2nd person has a 1/365 chance of sharing a birthday with the first person. Assuming they don’t, then the 3rd person has a 2/365 chance of sharing a birthday with either of the first two. The 4th person similarly has a 3/365 chance of sharing a birthday with any of the first 3 people. If you do all the math (which involves some stuff like flipping it into odds of *not* sharing a birthday and then taking the result away from 100%), you get to a >50% chance at 23 people.
Another way of looking at it is the number of pairs of people. When you have 2 people, you have 1 pair. When you have 3 people, you have 2 pairs. At 4 people, you have 6 pairs, and with 5 people you have 10 pairs. This keeps growing at an alarming rate. At 22 people you have 231 pairs, and at 23 people you have 253 pairs. While the odds of a single pair of people not sharing a birthday is >99%, if you multiply those odds together 253 times you get down to 49% chance. By the time you have 75 people in the room, there are 2775 combinations of people, so the odds drop to nearly 0.
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