Can you beat 50% guessing red in a deck?
Asked at Jane Street, Optiver
A standard shuffled deck ( red, black) is dealt face-up one card at a time. At any moment you may say "stop"; you then bet that the next card will be red, and win if it is. You may also let the deck run to the last card. You know the count of reds and blacks remaining at all times.
Is there a stopping strategy that gives you better than a chance of winning? Prove your answer.
Show a hint
At the moment you stop, what is the probability the next card is red? Try to relate any strategy to simply betting on the very last card in the deck.