Presentation issue:

Rather than expressing probabilities in reduced fraction form, it would make the argument easier to follow if groups of probabilities were expressed with common denominators.

So instead of

X:1/15, Y:1/30, A:3/5, B: 3/10

say

X:2/30, Y:1/30, A:18/30, B: 9/30

This makes it easy to compare the probablities using ratios of the numerators, with the denominators just there for normalization.