## Problem

You want to find someone whose birthday matches yours. What is the least number of strangers whose birthday you need to ask about to have a $$50$$-$$50$$ chance?

## Solution

Let $$X$$ be a random variable representing the number of people you need to ask before finding someone whose birthday matches yours. Then, applying the geometric distribution, we know $$P(X=n)$$ is:

$P(X=n) = \left(\frac{364}{365}\right)^n \left(\frac{1}{365}\right)$

Extending the above, the probability that $$X$$ is no more than some value $$n$$ is simply:

$P(X \leq n) = \sum_{i=0}^n \left(\frac{364}{365}\right)^n \left(\frac{1}{365}\right)$

To finish this problem, we assign $$P(X \leq n) = \frac{1}{2}$$ and solve for $$n$$:

$\frac{1}{2} = \sum_{i=0}^n \left(\frac{364}{365}\right)^n \left(\frac{1}{365}\right) = \frac{\frac{1}{365}\left[1-\left(\frac{364}{365}\right)^n \right]} {1-\frac{364}{365}}$

$\frac{1}{2} = \left(\frac{364}{365}\right)^n$

The first line above comes from applying the formula for the sum of a finite geometric series, which is $$s_n = \frac{a_1(1-r^n)}{1-r}$$. The final answer is $$\log_\frac{364}{365} \frac{1}{2} \approx 253$$.