First ace, with replacement or without
Asked at Jane Street
You draw cards from a standard -card deck looking for the first ace, one draw at a time.
Compare the expected number of draws to the first ace when you draw without replacement (an ordinary shuffle) versus with replacement (reshuffle after each draw).
Show a hint
Without replacement, use the gap/symmetry argument. With replacement, each draw is an independent trial, so the waiting time is geometric.
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.