Question: A dealer presents to you the following game: The dealer is going to deal cards from the top of a standard deck one card at a time and turn them over on a table for you to see. You can tell the dealer to stop dealing at any time. If the next card is red, you win. Assuming optimal play, what is the highest probability you can achieve of being correct?
I began by thinking that the optimal strategy is just to wait until $b-r = n$ where $b$ and $r$ are the number of black and red cards that have been drawn and then you work out $n$ * probability of getting $b-r=n$ at any point during the game but I am not sure how to do this. It seems too complicated though.
Another idea I had was that the highest probability that you are correct is always $\frac{1}{2}$ regardles of what strategy you use because the order of the cards is already determined beforehand with probability $\frac{1}{2}$ of a red or black card being drawn on any given position. But I guess this doesn't take into account the cards you have already drawn up until that position i.e. you need to take into account a conditional probability.
Any ideas?
