Conference rusure::math

1602.0. "Non-Transitive Dice" by BEING::EDP (Always mount a scratch monkey.) Mon May 04 1992 11:44

    This was in a Usenet thread on non-transitive dice -- three or more
    dice with numbers on their sides such that the probabilities of one
    dice rolling a greater number than another are not transitive.  At the
    bottom is apparently a best result for three dice.
    
    
    				-- edp
    
    
Article 27581 of sci.math:
Newsgroups: sci.math
Subject: Re: Non-transitive dice in Paulos' Innumeracy?
From: [email protected] (Arthur Rubin)
Organization: Beckman Instruments, Inc.
Lines: 34

In <[email protected]> [email protected] (USENET News System) writes:

>In article <a_rubin.704490201@dn66>, [email protected] (Arthur Rubin) writes:
>> 
>> >Yep, Martin Gardner did.  (I don't have my Scientific Americans available.)
>> 
>> D_1 beats D_2 with probability p
>> D_2 beats D_3 with probability p
>>  .
>>  .
>>  .
>> D_n beats D_1 with probability p
>> 
>> This, I can't answer.
>> --
>I have looked up the relevant Scientific Americans (December, 1970 and October, 1974).
>I can't answer for dice, but the following gives events D_1,..., D_n with the above
>property and p = (n-1)/n.

(details ommitted)

The critical property for dice is that the random variables D_i are independent.

For 3 dice, I believe the maximum is 1/GoldenRatio = 2/(1+5^(1/2)) = (5^(1/2)-1)/2 = g, with 
the following "dice".

D_1 :5 with probablility 1-g, 2 with probability g
D_2 :4 with probability g, 1 with probability 1-g
D_3 :always 3
--
Arthur L. Rubin: [email protected] (work) Beckman Instruments/Brea
[email protected] [email protected] [email protected] (personal)
My opinions are my own, and do not represent those of my employer.
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