I am faced with the following problem:
There are three four-sided dice $D_1$, $D_2$, $D_3$, $D_4$ that have the following assignments of numbers to their sides:
$D_1$: (1, 8, 11, 14)
$D_2$: (2, 5, 12, 15)
$D_3$: (3, 6, 9, 16)
$D_4$: (4, 7, 10, 13)
It is said that die $D_i$ beats die $D_j$ when the probability that die $D_i$ comes up with a larger number than die $D_j$ is greater than $1/2$ in an experiment where each of these two dice is rolled once.
Now, the trivial way to solve this problem is to iteratively go through each pair of die and determine how many outcomes result in $D_i$ being larger than $D_j$ and dividing that by the total number of outcomes to find the probability.
This seems tedious. Is there a better way to show the probability of each pair of die, given the different numbers on each side?