Having shuffled a deck of cards, I tell you that we will play a
game where I draw and reveal card after card from the top of
the deck. I want you to say "Stop!" at any time before the last
card has been drawn, and you win if the next card at that point
is red. What strategy should you use to maximize your chance
of succeeding?

To maximize your chance of drawing a red card after saying "Stop!", you should use a strategy based on the number of cards revealed and the number of red cards remaining. Here's a well-known strategy from probability theory called the "Optimal Stopping Theory" or "Secretary Problem":

1. **Divide the deck into two parts**: The first part should contain a certain fraction of the cards (commonly about \(1/e\) of the total number of cards), and the second part contains the rest.

2. **Observe the first part of the deck without saying "Stop!"**: In this phase, just count how many red cards have appeared. For a standard deck of 52 cards, the first part will be about 19 cards (since \(52/e \approx 19\)).

3. **Choose the next red card in the second part**: After you have gone through the first part, start saying "Stop!" the moment you encounter the next red card. This strategy relies on the principle that by sampling a certain number of cards, you get a good idea of the distribution, and you then have a higher probability of finding a red card in the remaining part of the deck.

### Steps:

1. **Divide** the deck of 52 cards. Approximately \(52/e \approx 19\) cards will be observed without stopping.
2. **Count** the number of red cards in the first 19 cards.
3. **Stop** at the first red card you encounter after these 19 cards.

By following this strategy, you maximize your probability of successfully stopping at a red card. 

Would you like more details on the mathematics behind this strategy or have any other questions?

Here are 5 more questions you might want to ask:
1. How does the Optimal Stopping Theory apply to real-life scenarios?
2. Can you explain the math behind the \(1/e\) rule in more detail?
3. What are some variations of the Optimal Stopping Theory?
4. How does this strategy compare to simply guessing?
5. What is the expected success rate using this strategy?

**Tip:** Understanding the underlying probability principles can often give you an edge in decision-making processes, both in games and real-life situations.

Discover the Optimal Stopping Strategy for maximizing your chances of drawing a red card from a shuffled deck. Learn about the mathematical concepts behind the strategy, including the Secretary Problem and the 1/e Rule, suitable for high school students and enthusiasts.

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