At a large convention of 1,000 people, what is the probability of having 10 people share the same birthday?
Thank you for help.

Guest Nov 15, 2018


Rom  Nov 15, 2018
edited by Rom  Nov 15, 2018

This answer is incorrect. You calculated the probability that EXACTLY 10 people share the same birthday for a given birthday, and you calculated that correctly, but when you multiplied the result by 365 to get the probability that there exists a birthday such that exactly 10 people share it you overcounted. Try your method for n and p that are small enough to calculate, you will see that sometimes you will get a probability that is larger than 1.


Besides, I think what the guest wants to calculate is the probability that for some birthday AT LEAST 10 people share it, not that exactly 10 people share it.

Guest Nov 15, 2018

There are actually several formulae for these "Birthday problems" depending on the number of people sharing the birthdays as follows:
Let d=365 =Number of days in a year.
Let n=Total number of people involved.
Let k =Number of people sharing same birthday

1 - Birthday probability for 2 people:
  p = 1 - ((d nPr n) / (d^n))

2 - Birthday probability for 3 people:
 p = 1 - ∑d!n! / {i! (n-2i)! (d-n +i )! 2^i d^n}, i=0 to n/2

3 - Birthday probability for 4 or MORE people.
 p = 1 - exp(-(d^(1 - k)* (exp(-n/(d* k))* n)^k)/((1 - n/(d* (1 + k)))* k!)).
So, for this last one, the problem becomes:
d =365, n=1,000, k=10 people (at least)
If you plug in these numbers into the formula above you should get: 0.186277 =~18.63%.
However, this last formula is supposed to give accurate results to within 0.5 of 1% up or down.

Guest Nov 15, 2018

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