Chuck-A-Luck, Bob Farmer

[Larry Barnowsky]

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% (125-91)/216, a real suckers bet.

[Bob Farmer]

Hi Larry, thank you for this excellent analysis. I'll think you'll find the next installment interesting because it may explain why your math professor was so adamant.

[Guest]

The one that got me ... in the game show with Monty Hall if he gives you the chance to switch doors, then your odds of winning actually improve if you do switch.

I understand it now, but it is counter intuitive. I guess if Paul Erdos had difficulty understanding it than I consider myself in good company :)

[Guest]

I just leafed through that article (Monty Hall Knows All) this morning. Having recently assisted my girlfriend in successfully completing Contemporary Mathematics and Statistics (I thought I was through with college in the eighties until she decided to go back and I somehow became an unwitting tutor!) my interest in odds and statistical anomalies has been rekindled. Especially as it relates to the old
"magicians choice", and methods to make those choices as fair and open in appearance as possible, while retaining overall control of the outcome.

Best,

Mike

[Bob Farmer]

Mike, the next column is a compilation of information on doing exactly what you want to do. I think you'll find the column and the references very useful.

[Guest]

Thanks for the heads up Bob. I'm looking forward to the July issue :)

Best,

Mike