Treebeard's Stumper Answer
Sackcloth and Ashes
Today is the Dunn Middle School Renaissance Faire! We have a village with peasants, craftsmen, royalty, minstrels, and more. You meet a gray-bearded penitent in sackcloth who offers a simple game of chance. He has three cards. One is black on both sides, one is red on both sides, and one has a red and a black side. You reach in his bag and pull a card to the table. It's black on top. The bearded man bets even money that the hidden side is the same. "It can't be red-red, so there are only two choices. We have an equal chance of winning." Is this a fair game or a con?
Here is that crusty gray-bearded penitent in sackcloth at the DMS RenFaire. This was a fun day!
I played my "game of chance" with three cards that I made by gluing some old playing cards together face-to-face and marking them like this on each side:
Show the cards to the player and shuffle them in the bag. Have the player reach in to pick a random card and put it down without looking at the bottom. Suppose the top is black. Either way, you bet that the bottom is the same color. Explain how that eliminates the red-red card, so it must be an honest wager since only two cards remain. As soon as you see their money, the player can turn the card for judgement. The game is especially vicious if you "double or nothing" every time you lose.
For added effect, you can reach into the bag and remove the red-red card (after the player draws black) to demonstrate that there are only two choices remaining. How does it effect the odds if you allow the player to switch cards at this point?
Wearing sackcloth and ashes is an ancient symbol of penance and humility. In my "game of chance", who should be mournful and why? What other games is this related to?
I wore sackcloth at the DMS Renaissance Faire, but I gave many others a reason for penitence with my simple card game. It seems fair. If your card is black on top, I can reach into the bag and remove the red-red card. Only two cards remain, but the odds are still 2-1 for me that your card is the same color on the bottom. Switching cards now gives you the advantage, but few kids took my offer. There's no sleight of hand. Our intuitions about probability can be dangerous. If it's still not obvious, make your own cards and play solitaire until you understand the con!
This is my busy time at Dunn Middle School, so I don't have time for a full analysis of my stumper game. I cleaned up at the DMS Renaissance Faire winning RenFair scrip and food. I know at least one student learned a hard lesson about progressive betting on what seems to be a fair game. Travis was the only student who figured out the real odds. I always have a 2-1 or 67% chance of winning with my bet that the bottom is the same color as the top. Removing a card from the bag doesn't change those odds.
My game is a simple form of several classic con or scam games including the Shell Game and Three Card Monty and the Monte Hall problem. These "games of chance" are all isomorphic, though the finesse of street players can and game show hosts can reduce your odds to zero.
I'll get back to this stumper as soon as I can with a real analysis. In the meantime, if it's still not obvious, you should make your own set of cards and play solitaire until you understand it. You can play a very similar game online at the Let's Make a Deal Applet (JAVA). Play for a while and keep track of your wins and loses.
Here are a few starting web links for your own research:
- I found this card game as the "Three-Card Swindle" in Martin Gardner's Aha! Gotcha: Paradoxes to Puzzle and Delight (Freeman, 1982), along with several more related probability gotchas. The DMS Renaissance Faire was the perfect chance to use this puzzle to teach a math lesson to my students. It's also discussed in Peter Mason's Half Your Luck (Penguin, 1986), A.K. Dewdney's 200% of Nothing (Wiley, 1996), and Julian L. Simon's Resampling text.
- Marilyn vos Savant was right about this one. (But sometimes Marilyn is wrong!) The "Monty Hall Problem" based on the Let's Make A Deal TV show has generated a huge amount of controvery, much of it wrong. By the math, you should always switch doors (or cards). But that bearded penitent might only give you the choice of switching when he knows you'll lose! There's more discussion here, here, here, here, and here.
- Junpei Sekino's "Who's the Goat . . . Marilyn or the Mathematicians?" from Willamette University explains the connection with the shell game and three-card monte. Of course there are three-card monte references in the Simpsons, e.g. Lisa the Beauty Queen [9F02] and The Springfield Connection [2F21]: "Three card monte! ... Woo hoo! Easy money!" There's more info on these classic con games at HowStuffWorks and here, here, and here.
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