Clue to Puzzle #14: 52 Cards Win a Dollar

14. You have 52 playing cards (26 red, 26 black). You draw cards one by one. A red card pays you a dollar. A black one fines you a dollar. You can stop any time you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing expected payoff?

Also, what is the expected payoff following this optimal rule?

This puzzle is really very hard. It's a puzzle though that you can make decent progress with long before getting a full answer. I cover this at length in the main answer page but for this hints page I'm going to breeze over that and go straight to giving clues to the final solution. Quant experts read on.

If you're going to solve this the following might be considered prerequisite.
  • We can't lose. We can always play to zero.
  • I'll quit when I get 'x' dollars pays the best return for x=4
  • Consider the case where I am one dollar up with one card left to draw. I should always quit here.
  • I'll quit when i get x dollars ahead, where x is static, is not an adequate rule.
Having established that, it is helpful to work on a smaller deck, a six card deck with three red cards and three black. The game can only be in 16 positions in total. (3 to 0 of either remaining.) At any one of those positions we can easily calculate our current cash position and the probability of harming or improving our position in the game.

At this point I'm dangerously close to giving too much away so feel free to stop. We essentially now have a cash position for each point in the game and the probabilities of moving to a better or worse position. We need to convert that into an expected return and compare the expected return from gambling to our current cash position.

The final clue, and feel free not to read on.... ....The expected return should be calculate as a probability weighting of the expected returns from the possible outcomes of the gamble. It's helpful to start at the end and work backwards.

Where next?
Questions Answer

© Nigel Coldwell 2004 -  – The questions on this site may be reproduced without further permission, I do not claim copyright over them. The answers are mine and may not be reproduced without my expressed prior consent. Please inquire using the link at the top of the page. Secure version of this page.

PayPal - The safer, easier way to pay online.
I always think it's arrogant to add a donate button, but it has been requested. If I help you get a job though, you could buy me a pint! - nigel

This Website Uses Cookies

To increase the functionality of the site. The cookies I apply do not uniquely identify you, by continuing to use this site you agree to let me place a cookie. I also have advert and analytics providers, my advertising provider (Google,) does provide personalised adverts unless you specify otherwise, with them. For more information click here.x