Probabilities

Each die is beaten by the previous die in the list, with a probability of 2/3:

${\displaystyle P(A>B)=P(B>C)=P(C>D)=P(D>A)={2 \over 3}}$

A conditional probability tree can be used to discern the probability with which C rolls higher than D.

B's value is constant; A beats it on 2/3 rolls because four of its six faces are higher.

Similarly, B beats C with a 2/3 probability because only two of C's faces are higher.

P(C>D) can be calculated by summing conditional probabilities for two events:

• C rolls 6 (probability 1/3); wins regardless of D (probability 1)
• C rolls 2 (probability 2/3); wins only if D rolls 1 (probability 1/2)

The total probability of win for C is therefore

${\displaystyle \left({1 \over 3}\times 1\right)+\left({2 \over 3}\times {1 \over 2}\right)={2 \over 3}}$

With a similar calculation, the probability of D winning over A is

${\displaystyle \left({1 \over 2}\times 1\right)+\left({1 \over 2}\times {1 \over 3}\right)={2 \over 3}}$

Best overall die

The four dice have unequal probabilities of beating a die chosen at random from the remaining three:

As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. The probability of die A beating C is 4/9 (A must roll 4 and C must roll 2). So the likelihood of A beating any other randomly selected die is:

${\displaystyle {1 \over 3}\times \left({2 \over 3}+{1 \over 3}+{4 \over 9}\right)={13 \over 27}}$

Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. The probability of die B beating D is 1/2 (only when D rolls 1). So the likelihood of B beating any other randomly selected die is:

${\displaystyle {1 \over 3}\times \left({2 \over 3}+{1 \over 3}+{1 \over 2}\right)={1 \over 2}}$

Die C beats D two-thirds of the time but beats B only one-third of the time. The probability of die C beating A is 5/9. So the likelihood of C beating any other randomly selected die is:

${\displaystyle {1 \over 3}\times \left({2 \over 3}+{1 \over 3}+{5 \over 9}\right)={14 \over 27}}$

Finally, die D beats A two-thirds of the time but beats C only one-third of the time. The probability of die D beating B is 1/2 (only when D rolls 5). So the likelihood of D beating any other randomly selected die is:

${\displaystyle {1 \over 3}\times \left({2 \over 3}+{1 \over 3}+{1 \over 2}\right)={1 \over 2}}$

Therefore, the best overall die is C with a probability of winning of 0.5185. C also rolls the highest average number in absolute terms, 3+1/3. (A's average is 2+2/3, while B's and D's are both 3.)

Variants with equal averages

Note that Efron's dice have different average rolls: the average roll of A is 8/3, while B and D each average 9/3, and C averages 10/3. The intransitive property depends on which faces are larger or smaller, but does not depend on the absolute magnitude of the faces. Hence one can find variants of Efron's dice where the odds of winning are unchanged, but all the dice have the same average roll. For example,

• A: 7, 7, 7, 7, 1, 1
• B: 5, 5, 5, 5, 5, 5
• C: 9, 9, 3, 3, 3, 3
• D: 8, 8, 8, 2, 2, 2

These variant dice are useful, e.g., to introduce students to different ways of comparing random variables (and how comparing only averages may overlook essential details).

Relation to Efron's dice

These dice are basically the same as Efron's dice, as each number of a series of successive numbers on a single die can all be replaced by the lowest number of the series and afterwards renumbering them.

• A: 01, 02, 16, 17, 18, 19 → 01, 01, 16, 16, 16, 16 → 0, 0, 4, 4, 4, 4
• B: 03, 04, 05, 20, 21, 22 → 03, 03, 03, 20, 20, 20 → 1, 1, 1, 5, 5, 5
• C: 06, 07, 08, 09, 23, 24 → 06, 06, 06, 06, 23, 23 → 2, 2, 2, 2, 6, 6
• D: 10, 11, 12, 13, 14, 15 → 10, 10, 10, 10, 10, 10 → 3, 3, 3, 3, 3, 3

Oskar dice

Oskar van Deventer introduced a set of seven dice (all faces with probability 1/6) as follows:^[7]

• A: 2, 02, 14, 14, 17, 17
• B: 7, 07, 10, 10, 16, 16
• C: 5, 05, 13, 13, 15, 15
• D: 3, 03, 09, 09, 21, 21
• E: 1, 01, 12, 12, 20, 20
• F: 6, 06, 08, 08, 19, 19
• G: 4, 04, 11, 11, 18, 18

One can verify that A beats {B,C,E}; B beats {C,D,F}; C beats {D,E,G}; D beats {A,E,F}; E beats {B,F,G}; F beats {A,C,G}; G beats {A,B,D}. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,

• G beats {A,B}; F beats {A,C}; G beats {A,D}; D beats {A,E}; D beats {A,F}; F beats {A,G};
• A beats {B,C}; G beats {B,D}; A beats {B,E}; E beats {B,F}; E beats {B,G};
• B beats {C,D}; A beats {C,E}; B beats {C,F}; F beats {C,G};
• C beats {D,E}; B beats {D,F}; C beats {D,G};
• D beats {E,F}; C beats {E,G};
• E beats {F,G}.

Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents' dice.

Grime dice

Dr. James Grime discovered a set of five dice as follows:^[8]

• A: 2, 2, 2, 7, 7, 7
• B: 1, 1, 6, 6, 6, 6
• C: 0, 5, 5, 5, 5, 5
• D: 4, 4, 4, 4, 4, 9
• E: 3, 3, 3, 3, 8, 8

One can verify that, when the game is played with one set of Grime dice:

• A beats B beats C beats D beats E beats A (first chain);
• A beats C beats E beats B beats D beats A (second chain).

However, when the game is played with two such sets, then the first chain remains the same (with one exception discussed later) but the second chain is reversed (i.e. A beats D beats B beats E beats C beats A). Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option):

Sets chosen by opponents		Winning set of dice
		Type	Number
A	B	E	1
A	C	E	2
A	D	C	2
A	E	D	1
B	C	A	1
B	D	A	2
B	E	D	2
C	D	B	1
C	E	B	2
D	E	C	1

There are two major issues with this set, however. The first one is that in the two-die option of the game, the first chain should stay exactly the same in order to make the game intransitive. In practice, though, D actually beats C. The second problem is that the third player would have to be allowed to choose between the one-die option and the two-die option – which may be seen as unfair to other players.