Expected Payoff of the Red-Black Card Game

Expectation · Hard · Free problem
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?

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