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 50-50 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 $\left(\frac{N-r}{N}\right)^\frac{1}{p} p$.
Since we want to find the number that gives us a 50-50 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:
\[f(x) = \begin{cases} 1-\left(\frac{365}{364}\right)^{-\lfloor x\rfloor-1} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases}\]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”. ↩︎