Choose three length-3 patterns A, B, C over {H,T}, and give each its own biased coin: pA=P(H on coin A), pB=P(H on coin B), pC=P(H on coin C). In each run we flip the three coins independently until each pattern appears; then we compare who “wins first”. Pairwise wins can form non-transitive cycles: A beats B, B beats C, yet C beats A (with random tie-breaks).
McDonald-shaped region: drag to rotate, scroll to zoom, click to select (pA,pB,pC).
Sampling: ready.
Region definition: 0<pA,pB,pC<1 and G_AB(pA,pB)>0, G_BC(pB,pC)>0, G_CA(pC,pA)>0. (These polynomials correspond to the classic length-3 example A=HHH, B=HTH, C=HHT.)
Run N Monte Carlo games at the selected (p,q).
Runtime: ready.
Means are exact; pairwise win probabilities are simulation estimates.
Counts based on first occurrence times in each run.
| Comparison | Wins | Estimated P |
|---|---|---|
| A < B | — | — |
| B < C | — | — |
| C < A | — | — |
| Cycle frequency | — | — |
Each run flips 3 independent coins (A/B/C) until all three patterns have appeared, then records the three pairwise outcomes (with random tie-breaks).