Fig 8
(click on Figures 8 and 9 to see an interactive version)
The lowest values for the probability of non-transitive cycles are \(\{\) \(\frac{18}{35}\), \(\frac{17}{33}\), \(\frac{16}{31}\), \(\frac{15}{29}\), \(\frac{14}{27}\), \(\frac{13}{25}\), \(\frac{12}{23}\), \(\frac{11}{21}\), \(\frac{10}{19}\), \(\frac{19}{36}\), \(\frac{9}{17}\), \(\frac{17}{32}\), \(\frac{8}{15}\), \(\frac{15}{28}\), \(\frac{7}{13}\), \(\frac{13}{24}\), \(\frac{19}{35}\), \(\frac{6}{11}\), \(\frac{17}{31}\), \(\frac{16}{29}\), \(\frac{5}{9}\), \(\frac{19}{34}\), \(\frac{9}{16}\), \(\frac{17}{30}\), \(\frac{4}{7}\), \(\frac{15}{26}\), \(\frac{7}{12}\) \(\}\) and they vary from \(\frac{18}{35}\) to \(\frac{7}{12}\), appearing with different frequencies as shown in Figure 8.
To get an idea not only of the lowest values for the probabilities of each non-transitive cycle but also of the three probabilities associated to each cycle, we can make a graphical representation of those triplets of probabilities. Note that the representation of a triplet by a point involves the (arbitrary) choice of the firts element of the triplet. The other choices correspond to a rotation of \(120^{\circ}\) and \(240^{\circ}\), about the right line through the origin with the direction defined by the vector \((1,1,1)\). Those three sets of points are represented in different colours in figure 9. In Figure 10 the stereoscopic pair represents a similar set of probabilities but corresponding to cycles with three dice with 1 to 4 spots on each side.
