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Confusing Dice Game

Two players are needed for this experiment.

Create three "fair" dice: red (R), green (G) and white (W), with their sides marked as follows:

R: 1, 1, 4, 4, 5, 5
G: 3, 3, 3, 3, 3, 3
W: 2, 2, 2, 2, 6, 6

Each of the two players picks and rolls a die. The high roll wins that turn. Each player rolls their die 30 times, then each player picks a different die and then repeat the process.

Record the outcomes using tally marks on this chart:

R beats G:
G beats W:
W beats R:
G beats R:
W beats G:
R beats W:

If you were to play this game to WIN (i.e. for money), which die would you select? Why?


Hint: In order to discover that the situation becomes quite confusing (by intention), you need to make sure you have played the games an ample number of times (e.g. 30 times for each pair RG, RW, and GW) and recorded your tallies.

Given a pair of dice (e.g. RG, RW, or GW), which die tends to win? What is the "surprise"?


Solution Commentary: After playing the games, one should discover that since G beats W two-thirds of the time and W beats R five-ninths of the time, you would expect that G would beat R....BUT R beats G two-thirds of the time!

How can this happen? Draw a tree model to illustrate and document this surprising result.

If you want to learn more about this confusing situation, consult "non-transitive dice"...checking out Efron's four dice and Miwin's dice.