Given a deck of 52 card, 26 black and 26 red, the player pulls cards one by one, seeing each pulled card's color. At any moment the player can stop and pull the final card. If this card is red, he wins, he loses otherwise.
Since everything is finite and discrete, I considered using dynamic programming to brute-force this task, calculating the mean of an indicator variable of winning for each $r$ and $b$, the amount of red and black cards pulled, respectively. Stopping if the expected value in the current moment is higher than the expected value after pulling the next cards seems to be the optimal solution, but I couldn't prove that rigorously. Is there a more elegant solution than brute force? A proof for my solution would be also appreciated.
You can prove by induction that the value of the game is always the current proportion of red cards, and you obtain this value whether you stop or continue, so it doesn’t matter when you stop.