Thats probably a title guaranteed to make readers close down their browsers or frantically hit the back button, but this is interesting – honest. Here’s the question. How many people would need to be in a room, for it to be more likely than less likely that two of them share a birthday?
The answer may surprise you. It’s 23.
Here’s how to work it out. It’s actually easier to work out the probabilities that the people don’t share a birthday. So, when there are two people in the room, the probability that they don’t share the same birthday is 364/365, (or 0.99726) In other words, almost certain.

When the third person enters, the probability that the three don’t share the birthday is (364/365) x (363/365) = 0.9918. Still very very likely.

If we carry on in this way, adding one person to the room at a time, the probabilities work out as shown below (i’ve missed out the calculations in middle to save time, but you should be able to see the pattern by now.)
4th person – (364/365) x (363/365) x (362/365) = 0.9836
5th person – (364/365) x (363/365) x (362/365) x (361/365) = 0.9729
22nd person – (364/365) x (363/365) x (362/365) x (361/365) x (360/365) x (359/365) x (358/365) x (357/365) x (356/365) x (355/365) x (354/365) x (353/365) x (352/365) x (351/365) x (350/365) x (349/365) x (348/365) x (347/365) x (346/365) x (345/365) x (344/365) = 0.5243 – we’re nearly there, thank god for Google calculator
23rd person – (364/365) x (363/365) x (362/365) x (361/365) x (360/365) x (359/365) x (358/365) x (357/365) x (356/365) x (355/365) x (354/365) x (353/365) x (352/365) x (351/365) x (350/365) x (349/365) x (348/365) x (347/365) x (346/365) x (345/365) x (344/365) x (343/365) = 0.4927
There we are, the probability has dropped below 0.5 for the first time, so this is the point where it is more likely than less likely that two people share a birthday. Told you it was interesting….ish.

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