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Various versions of this paradox have appeared over the years.  For this one the probabilities are as follows:

Pr(A outscores B) = 5/9
Pr(B outscores C) = 5/9
Pr(A outscores C) = 4/9

i.e. A is better than B i.e. B is better than C i.e. A is NOT better than C

So, over a large number of trials, in spite of the fact that spinner A beats spinner B and B beats C, spinner A does not beat spinner C! Thus the binary relation outscores is NON-TRANSITIVE which is a counter-intuitive result.
Green (1981) has described a set of three non-transitive dice in an article published in the journal Mathematics in School, where they are referred to as Chinese Dice.  These dice are marked as follows:

Die A     6, 6, 2, 2, 2, 2
Die B     5, 5, 5, 5, 1, 1
Die C     4, 4, 4, 3, 3, 3

For these:

Pr(A outscores B) = 5/9
Pr(B outscores C) = 6/9
Pr(A outscores C) = 3/9

(Note that these three probabilities total to 14/9 as do those for the spinner set.)
Following the publication of that article, various people wrote with information, including Ainley who gave details of a four dice set which he refers to in his book Mathematical Puzzles (Ainley, 1978).  Those dice are marked thus:
 

Die A     7, 7, 7, 7, 1, 1
Die B     6, 6, 5, 5, 4, 4
Die C     9, 9, 3, 3, 3, 3
Die D     8, 8, 8, 2, 2, 2

These have the pleasing property that each pair taken cyclically gives the same probability:

Pr(A outscores B) = 2/3
Pr(B outscores C) = 2/3
Pr(C outscores D) = 2/3
Pr(D outscores A) = 2/3

(This property holds for our three-spinner set too, but not for Green's three dice set.)
A different four dice set is used in Practical Statistics (Rouncefield and Holmes, 1989).  Those are marked:

Die A     2, 2, 2, 2, 6, 6
Die B     5, 5, 5, 1, 1, 1
Die C     4, 4, 4, 4, 0, 0
Die D     3, 3, 3, 3, 3, 3

This set, which is discussed in a book by Hensberger (1979), is in fact equivalent to Ainley's set, as can be shown by adjusting the numbers while still preserving the 'outscores' relation.  The transformation required to convert the first set to the second is:

9 goes to 6, 8 goes to 5, 7 goes to 4, 6 goes to 3, 5 goes to 3, 4 goes to 3, 3 goes to 2, 2goes to 1, l goes to 0.

Taking matters a stage further, Ainley has written to say that the "goodness" of diceis transitive if the winner is paid by the loser an amount proportional to the difference between the numbers thrown, rather than receiving a fixed stake.  In that case the "goodness" of each die can be represented by the total of its face values.  For example, in the set of spinners in this article, each spinner has a total face value of 15 indicating that each die is as "good" as the others if the payout is proportional to the difference in the scores.  It can be confirmed by simple probability calculations that the expected winnings for A when A plays B is

{(1-3) + (1-4) + (1-8) + (5-3) + (5-4) + (5-8) + (9-3) + (9-4) + (9-8) 1 }/9= 0.

Similar calculations for when B plays C and when C plays A also yield zero.  For the set of dice described by Green (1981) the total face values differ and the expected winnings are not zero.  Investigating why the total face value is a measure of the "goodness" of a die is an interesting exercise.

We would like to hear of any other essentially different sets or of other interesting properties that such sets have and we would like to receive suggestions for investigations on this or other topics. Teaching Statistics would like to publish details of any good ideas for Practical Activities that are sent in.

References
Ainley, S. (1978).  Mathematical Puzzles, Bell and Hyman.
Green, D.R. (1981).  How Probability Pays,  Mathematics in School, 10(2), 23-24.
Hensberger, R. (1979).  Mathematical Plums. Dolciani Mathematical Expositions, 4. The Mathematical Association of America..
Rouncefield, M. and Holmes, P. (1989). Practical Statistics, Macmillan.
 

In the book Teaching Statistics at its Best this article is followed by a description of a game to decide Which Spinner is the Winner? with practical instructions and tables for results. The worksheet Which Spinner is the Winner? includes various discussion points for consideration when looking at the results of experiments involving spinners.

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Please email: alison.davies2@ntu.ac.uk with any comments or corrections.

© The Teaching Statistics Trust 2006. The Teaching Statistics Trust is a registered charity.
ISSN 0141-982X (Print) ISSN 1467-9639 (Online)