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
The best way to think about it is to first realize that when comparing birthdays for 23 people you’re not just making 22 comparisons, you’re making 253.
Why’s that? Because you first compare Person 1 to the other 22 people, that gives you 22 comparisons. You then remove Person 1 and compare Person 2 to the other 21 people remaining, that gives you another 21 comparisons. You then remove Person 2 and compare Person 3 to the 20 people remaining, that gives you 20 more comparisons. You continue this until you’ve compared the birthdays of all 23 people with each other. 22+21+20+19….+3+2+1 = 253
This means that in order for two people to not share a birthday, ALL 253 comparisons need to have no matches. The odds of a single comparison not being a match are 364/365 = 0.99726027 or 99.72%. If you’re making 253 comparisons then the odds of every one of those not matching are (0.99726027)^253 which is 0.4995 or 49.95%. If the odds of no matches between 23 people are 49.95% that means that the odds of at least 1 match are 50.05%.
Ultimately, the reason the birthday paradox doesn’t makes sense at first glance is because people are assuming you’re only making 22 comparisons but when you really lay it out you realize that there are actually 253 total comparisons.
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