The EV to drawing another card is going to be tied to the probabilities of continued random walks eventually landing on higher scores than you currently have. You stop as soon as the EV is negative (or 0 if you’re not a gambler).

That calculation is presumably the tricky part… integrating over Pascal’s Triangle or something. But it must be positive to start: If the first card is red we have already gained, and if it’s black we’re at worst back where we started.