Looking back, I think I got my numbers mixed up (thanks, beer!). $12 or $13 is probably the number I'm looking for, not $26.

Anyway, my logic is if I am at or higher than the median after I have drawn half the deck, then that means there are more black cards than reds in the remaining deck and I have a greater chance each draw of losing a buck than gaining.

So it is in my best interest to pull out at that point, than to try my luck against ever worsening odds.

For the same reason, you’d never intentionally stop at $0 (unless the deck is exhausted). You can always get back to $0 by drawing the rest of the deck, and there’s a 50% chance of going up a dollar on the next card.

If your sum is currently positive, then the chance of going up a dollar is below 50%.