I have set up a stall where you may play a fabulous game.
 
Presented before you are 4 unusual 6-sided dice:
 

A Big Red die:
has the numbers 5, 5, 5, 5, 1 and 1 on its sides

A Large Yellow die:
has the numbers 6, 6, 2, 2, 2 and 2 on its sides

A Medium Green die:
has the numbers 6, 4, 4, 2, 2 and 1 on its sides

A Small Blue die:
has the numbers 3, 3, 3, 3, 3 and 1 on its sides
 
I inform those that are unaware that the average value they would roll with each of the 4 dice are (roughly) 3.66, 3.33, 3.17 and 2.67 respectively. All dice are fair and players find it impossible to cheat when rolling them.
 
I request a $1 payment from you to play. You may choose any one die. Then I may choose any of the remaining dice. We then roll. If you roll more than or the same as me, I return your original $1 stake and a bonus $1 prize. If I score more than you, I keep your stake and you win nothing.
 
What would be your strategy if you wanted to walk away from my stall with the most amount of money possible?
 
(Thanks go to an old university professor would showed us something similar, which instantly intrigued me, in a Probability lecture)

First set up a win/loss table for each pair of dice.

The table for the red/yellow pairing would look like this:

  5 5 5 5 1 1
6 Y Y Y Y Y Y
6 Y Y Y Y Y Y
2 R R R R Y Y
2 R R R R Y Y
2 R R R R Y Y
2 R R R R Y Y

There are 16 wins for red, and 20 for yellow and no ties Express this as R/Y=(16, 20, 0)

The pairings are:

R/Y = (16, 20, 0)
R/G = (20, 14, 2)
R/B = (24, 10, 2)
Y/G = (14, 12, 10)
Y/B = (16, 20, 0)
G/B = (20, 15, 1)

Once you have chosen your die, the dealer will choose the die that gives him the greatest number of wins against that die

If you choose red, he'll choose yellow => Y/R = (20, 16, 0)
If you choose yellow, he'll choose blue => B/Y = (20, 16, 0)
If you choose green, he'll choose red => R/G = (20, 14, 2)
If you choose blue, he'll choose green => G/B = (20, 15, 1)

So, no matter what you choose, your odds are 16:20, or 4:5, and (on average)for every nine rolls, you will lose a dollar.

Posted by TomM on 2002-07-24 03:44:02