[nramanand]

Not sure I followed what you meant by >= $26 in 2a. The max profit one can make is $26, by drawing 26 red cards in a row, never more.

You are correct though that you'd never want to stop on a negative number, as you always can execute 3b

[Dalewyn]

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.

[chrchang523]

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.

[Filligree]

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