The game
A standard deck is shuffled and dealt face up one card at a time. At any point you may say "stop" and bet that the next card is red. You must bet before the deck runs out. What is your best strategy, and what is your win probability?
The answer
1/2, under every strategy. There is nothing you can do.
Why
Let the deck have r red and b black cards remaining. The probability the next card is red is r/(r + b).
Now consider your expected win probability if you decline to bet and instead wait. With probability r/(r+b) the next card is red, leaving r-1 red; with probability b/(r+b) it is black, leaving r red. The expected value of the next proportion is
(r/(r+b)) times (r-1)/(r+b-1) + (b/(r+b)) times r/(r+b-1)
which simplifies to exactly r/(r+b).
The proportion of red remaining is a martingale: waiting neither improves nor worsens your position, on average. By optional stopping, any stopping rule you choose has the same expected outcome as betting immediately - which is 26/52 = 1/2.
The intuition
You are tempted to wait for a run of black cards, reasoning that red must now be richer. That is true - but the probability you get to see such a run exactly offsets the improved odds when you do. The deck is not obliged to give you a good spot, and the times it does not, you are forced to bet in a worse one.
Why this is asked
It is a direct test of whether you will invent an edge that does not exist. Candidates often produce elaborate stopping rules and confident stories about why they work. On a trading desk that instinct is expensive, and interviewers know it.
The strong answer names the martingale, shows the one-step calculation, and says plainly that no strategy helps.
The variant that does have an answer
You may bet that the next card is red, and you get to choose the stake. Now Kelly applies and the interesting question is sizing rather than timing.
You are told the last card is red. Then the final card is a certainty and the answer changes completely - worth checking whether the question has slipped in a condition like this.
See martingales.