Expected Payoff of the Red-Black Card Game
A deck has 50 red cards and 50 black cards (100 total). You draw one card at a time without replacement. After each draw, you can choose to stop or continue. If you stop after drawing $k$ cards, your payoff is:
$\text{Payoff} = \frac{\text{number of red cards drawn}}{k} \times \
$
If you never stop, you draw all 100 cards. What is the expected payoff of this game under optimal play?