Question 39
Rosencrantz and Guildenstern each start with 2013 dollars and are flipping a fair coin. When the coin comes up heads Rosencrantz pays Guildenstern 1 dollar and when the coin comes up tails Guildenstern pays Rosencrantz 1 dollar. Let $f(n)$ be the number of dollars Rosencrantz is ahead of his starting amount after $n$ flips. Compute the expected value of $\max \{f(0), f(1), f(2), \ldots, f(2013)\}$.