Flip cards in a shuffled deck. Win double if red, lose half if black. You can choose whether to bet or not each time. What’s the optimal worst case scenario strategy? Start by considering a situation where we don’t bet until the final card or two and work backwards from there.
So far I've worked out that with $r$ red cards left, $n$ total cards left and $n-r$ red cards left:
$E[n, r] = E[n-1|r] \frac{n-r}{n} + E[n-1|r-1] \frac{r}{n}$
A bit stuck here now, I figure you won't bet if the above expectation is lower than your current winnings, which is going to happen if there are more black than red cards left. Not sure how to formalise it in such a way to find a closed solution though.