Finding Your Birthmate
21 Jul 2015
You want to find someone whose birthday matches yours. What is the least number of strangers whose birthdays you need to ask about to have a 5050 chance? ^{1}
The problem can be modeled as a negative binomial with parameters $r = 1$ and $p = 1/365$, hence the PMF is given by $\left(\frac{Nr}{N}\right)^\frac{1}{p} p$.
Since we want to find the number that gives us a 5050 chance, we need to calculate the inverse CDF of the negative binomial and find the value for $0.5$. The CDF of the negative binomial is given by:
I still don’t know how to compute the inverse of this CDF, but a numerical approximation gives us 252. In Maxima, one can use: quantile_negative_binomial(0.5, 1, 1/365);

This is problem 32 of Frederick Mosteller’s “Fifty Challenging Problems in Probability”. ↩