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I know that the Birthday Paradox is the fact that in a room of 23 people, the chances are more than 50 percent that at least two people share a birthday. However, this is under the assumption that all the birthdays are equally likely. My question is, if there is some variability in the probability distribution of birthdays, will 23 people still be enough to get more than 50 percent? Note, I am not saying that 23 people will be the least such number, but simply that it is greater than or equal to the least such number for any possible probability distribution of birthdays.

user107952
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  • The probabilities will certainly change (e.g. if there is a 100% chance that a random person is born on 1 January, then there is a 100% chance that any two people in a room will share a birthday, hence only two people are needed to exceed 50%), but the precise way in which they change is going to depend very much on the probability distribution of birthdays. – Xander Henderson Jul 17 '23 at 19:10
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    Assuming that birthdays are independent, but not uniformly distributed, then the probability of at least one match increases compared to a uniform distribution with the same number of people. For some distributions, the number needed for the probability of at least one match to exceed $50%$ will be lower than $23$; for none of them will it exceed $23$. (You could include $366$ day years too, which might push the probability down a little, but for $23$ people would still exceed $50%$) – Henry Jul 17 '23 at 19:11

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