this post was submitted on 02 May 2024
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The birthday paradox derives from how the chance of somebody there having their birthday on a specific day is 1-in-365 (ish)/nr-of-people hence the chance of two people having their birthday on that specific day is 1-in-365^2/nr-of-people, but the chance of two people having their birthday in the same day out of any days of the year is quite different because it's not a specific day anymore so it's quite a different calculation (which I totally forgot ;)).
~~In here the closest to that paradox would the chance of 2 whistleblowers of any company with whistleblowers dying within a few weeks of each other (which, depending on how many companies have whistleblowers, can be quite high) compared to the chance of 2 whistleblowers of Boeing dying within a few weeks of each other (which is statistically a lot lower unless there are thousands of Boeing whistleblowers).~~
Edit: actually it's more the chance of any 2 Boeing whistleblowers dying with a few weeks of each other at any point in time (so this includes long after they did it) vs the chance of any 2 Boeing whistleblowers dying with a few weeks of each other during the time they are blowing the whilstle.
The probability of 2 people having the same birthday is 1 in 365 because it's the same as picking person A's birthday as a specific day in the year and checking whether person B has their birthday on that date.
Now, the reason the number is so low is that you are basically comparing pairs and with 23 people there are 253 different pairings (23 choose 2 or
22*23/2). With each pair having a 1/365 chance to have the same birthday and having 253 distinct pairs, you would have to fail a 1/365 check 253 times in a row. The formula you can use for the success rate is1 - (1-p)^xwith p being the probability and x the number of trials, so in this case1 - (1 - 1/365)^253 = 0.5004In essence, the unintuitive part of the "paradox" is how fast the number of possible pairs grows the more people you add.