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?
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 \(\log_\frac{364}{365} \frac{1}{2} \approx 253\).