I thought it would be fun to share a little math
with you,
so I wrote up a
page how dice A can beat dice B, while B beats C, and C beats A.
You walk into a bar with three (three sided) dice A,
B, C, and find a man to gamble with.
For concreteness,

1^{st} side 
2^{nd} side 
3^{rd} side 
Dice A 
10 
21 
32 
Dice B 
11 
22 
30 
Dice C 
12 
20 
31 
You will each roll one dice. The one with the lower number pays the other $10.
To be fair, you let him reach roll choose his favourite dice.
If he chooses A, you choose B.
If he chooses B, you choose C.
If he chooses C, you choose A.
Is it possible that you always expect to win 5 out of every 9 rolls?
So the expected/average gain is $10´5/9 $10´4/9 = $1.11 each roll. Not a big profit.
If he has A and you have B 
If he has B and you have C 
If he has C and you have A 




An explanation is the following.
·
We each roll a dice. Let “Blue” be the event that your
10’s digit is different than his.
See all the blue Lose/Win.
This happens 2/3 = 6/9 of the time.
Because whatever his first digit is, you
have 2 out of 3 digits that are different.
·
Conditioned on “Blue”, the probability of you
winning is ½.
Note in 3 out of the 6 blue cases, you win.
The reason is that there is symmetry between you and him winning.
We write Pr[You winBlue] = ½.
·
Conditioned on “Red”, the probability of you winning is 2/3.
Note in 2 out of the 3 red cases, you win.
The reason is that if you count around 0,1,2,0,1,2,0,....
your second digit (in the red case) is always one ahead of his.
Eg He gets 10 on dice A, you get 11 on dice B. 0 vs 1
Eg He gets 21 on dice A, you get 22 on dice B. 1 vs 2
Eg He gets 32 on dice A, you get 30 on dice B. 2 vs 0
And you win 2 out of three of these.
We write Pr[You winRed] =2/3.
·
Combining these we write
Pr[You win] = Pr[Blue] ´ Pr[You winBlue] + Pr[Red] ´ Pr[You winRed]
= 2/3
´ ½ + 1/3
´ 2/3
= 3/9
(note 3 blue wins) + 2/9
(note 2 red wins)
= 5/9 (note 5 wins out of 9)
A general principle is
that in most fair games,
if your opponent chooses his strategy first and reveals it to you,
then you can respond with a strategy that beats his.
This is why to win you need secrecy.