Question
You have a devious older brother that wants to antagonize you. You are trying to make breakfast, and there are $N \geq 1$ fresh bagels and $N$ expired bagels. You can arrange them into two boxes however you would like. Afterwards, your brother selects a box at random and a bagel uniformly at random from the selected box. Afterwards, he takes the rest away. Suppose you are hungry and really like fresh bagels, so you want to maximize the probability of your brother drawing out a fresh bagel. Under an optimal strategy, what is the probability of your brother drawing a fresh bagel when $N = 100$?