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Larry Barnowsky Inner circle Cooperstown, NY where bats are made from 4770 Posts |
In the June issue of Genii Bob Farmer mentions the casino game Chuck-a-Luck and how the real odds are worse than one would figure from just common sense. I wrote a response to this in the Genii forum and decided to list it here as well.
The game is played with 3 dice. The player chooses a number (say six). Three dice are thrown. If a six comes up the player wins at even odds. If two sixes come up they get 2 to 1 odds, and if three sixes come up they get paid 3 to one. "The apparent odds in Chuck-A-Luck are very counterintuitive. When I was in high school I showed this game to a very smart math teacher. He couldn't accept that the odds actually favored the house. We then played the game for an hour with the house of course winning and he still thought this was a trick since he knew I did magic. The reason it works is the probability of not throwing a given number (say six) is (5/6)(5/6)(5/6)= 125/216 (house wins) 57.8% The probability of throwing one six is 3(5/6)(5/6)(1/6)= 75/216 The probability of throwing 2 sixes is 3(1/6)(1/6)(5/6)= 15/216 The probability of throwing 3 sixes is (1/6)(1/6)(1/6)=1/216 If the game is played 216 times the house will win at 1:1 odds $125 The player will win at 1:1 odds $75, at 2:1 odds $30 {15 times 2), and at 3:1 odds $3, for a total player winnings of $108 compared to the $125 the house won. The house makes $17 on every $216 bet for an edge of 7.87%. It's interesting that you can come with the same percentage by cubing 5/6. That's because the double and triple odds bring the payoff up to 108 which is half of 216 (instead of 91). Without the double and triple odds the edge to the house is 15.7% (75+15+1)/216, a real suckers bet." The last line should have read {125-(75+15+1)}/216 |
Scott Cram Inner circle 2678 Posts |
Here's another way to tell how good the odds are for the house. Walk into the Tropicana Hotel and Casino in Las Vegas, and two Chuck-a-Luck tables are the first thing you'll see.
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DaveS Veteran user New York 329 Posts |
What makes this game so seductive is the player's feeling that he has the "edge." (i.e., three chances for his number to come up + the possibility of double/triple pay-offs). Of course, the house's edge lies in those doubles and triples, which is what makes the spinning wheel versions, with its many double/triple combinations, so addictive. As I recall. this was a popular fundraising activity used by religious organizations at bazaars when I was growing up in NYC. Diabolical!
DaveS
We shall not cease from exploration/And the end of all our exploring/Will be to arrive where we started/And know the place for the first time. (TS Elliot)
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772ark New user 5 Posts |
I do enjoy playing Chuck-a-Luck, although I'm usually the dealer.
Never be afraid to try something new. Remember that a lone amateur built the Ark. A large group of professionals built the Titanic.
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JasonEngland V.I.P. Las Vegas, NV 1728 Posts |
Just to underscore how lopsided Chuck-A-Luck (originally called "Sweat Cloth and played as Crown and Anchor in the UK) is here are what the payoff odds would have to be to make the game a fair one:
1 number: 1 to 1 2 numbers: 2 to 1 3 numbers: 20 to 1 You could also pay 1:1, 3:1 and 5:1 and have a fair game. Jason
Eternal damnation awaits anyone who questions God's unconditional love. --Bill Hicks
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Scott Cram Inner circle 2678 Posts |
Martin Gardner has a good brief write-up about chuck-a-luck in Aha! Gotcha!.
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federico luduena Loyal user Spain 248 Posts |
Here's Galileo's letter to the Grand Duke of Tuscany, where Galileo constructs a probability table for three dice:
http://www.leidenuniv.nl/fsw/verduin/stathist/galileo.htm Even with this clear help I have difficulty reasoning properly re: Chuck-a-luck odds (and many others, for that matter). |
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