You flip a coin with the goal of maximizing the ratio of heads to total flips; you can decide to stop whenever. What is the expected value of this ratio?
My thoughts: The ratio is at least $3/4$. With $1/2$ probability we flip heads first and stop.
Otherwise, we flip until the ratio is $1/2$ (random walks return to the origin with probability $1$) giving an expected ratio of $3/4$. Can we do better?
