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If I have 5000 Facebook friends, what is the probability that a day with no one having that birthday exists? I assume there are 365 days in a year, and a uniform distribution of the dates of birth.

I guess that it's easier to calculate the opposite, and to subtract it from one.

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    a year is 365 days, and a uniform repartition of the births2011-09-14
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    Off topic: If you have 5000 FB friends, you have bigger problems :)2011-09-14
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    @percusse: I guess the OP wants to know how big his problems are, by checking if he has at least one day each year that he doesn't have to go to a birthday party :)2011-09-14
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    I would say that the result is 1-C(5000, 365) which is basically 0.2011-09-14
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    If by $C(5000,365)$ you mean the binomial coefficient "5000 choose 365", which is a big integer, then $1-C(5000,365)$ is a big negative integer, which is neither a probability nor "basically 0".2011-09-14
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    i forgot to devide by the total amount of possibilities right. It's late2011-09-14
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    Do we have to take in account if someone's birthday is on the 29th of feburary and we are not in a leap year?2011-09-14
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    A formal expression for the probability can be written using the multinomial coefficient: $$1-\frac{1}{365^{5000}}\sum_{d}{5000\choose d_1,\cdots,d_{365}},$$ where the sum is over indices $d_i\ge1$ with $d_1+\cdots+d_{365}=5000$. (This ignores February 29th however.)2011-09-14
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    The probabilities are dramatically different if you include leap years, see my answer.2011-09-15

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