Given two alternatively numbered dice:
Die A: $2\ 2\ 4\ 4\ 5\ 6$
Die B: $3\ 3\ 3\ 4\ 4\ 5$
How can I calculate the probability that Die A will roll higher than Die B for this pair or any other pair of alternatively numbered dice?
Obviously, I can just count the pairs of possible rolls and compare the numbers (This method gives me a $5/9$ chance for Die A to roll higher). However, this method seems slow as I would be generating and comparing a large number of dice as a means to check if a set of dice is nontransitive. So how would I find the probability analytically?
I cannot think of a method that would not involve counting faces on the die. However, you can choose methods which do so efficiently.
$$\def\P{\mathop{\sf P}}\begin{align}\P(A>B)&=\P(A>3)\P(B=3)+\P(A>4)\P(B=4)+\P(A>5)\P(B=5)\\&=\tfrac 46\tfrac 36+\tfrac 26\tfrac 26+\tfrac 16\tfrac 16\\&=\tfrac{17}{36}\end{align}$$