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?