Efron's Dice
A set of four nontransitive Dice such that the probabilities of A winning against B, B against C, C against D,and D against A are all 2:1. A set in which ties may occur, in which case the Dice are rolled again, which givesOdds of 11:6 is
References
Gardner, M. ``Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference.'' Sci. Amer. 223, 110-114, Dec. 1970. Honsberger, R. ``Some Surprises in Probability.'' Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 94-97, 1979.
- Hardy-Ramanujan Number
- Hardy-Ramanujan Theorem
- Hardy's Inequality
- Hardy's Rule
- Harmonic Addition Theorem
- Harmonic Analysis
- Harmonic Brick
- Harmonic Conjugate Function
- Harmonic Conjugate Points
- Harmonic Coordinates
- Harmonic Decomposition
- Harmonic Divisor Number
- Harmonic Equation
- Harmonic Function
- Harmonic-Geometric Mean
- Harmonic Homology
- Harmonic Logarithm
- Harmonic Map
- Harmonic Mean
- Harmonic Mean Index
- Harmonic Number
- Harmonic Progression
- Harmonic Range
- Harmonic Ratio
- Harmonic Segment