So we multiply prior proportions of $~$1 : 4$~$ for red vs\. blue by relative likelihoods of $~$3 : 1$~$ and end up with final proportions of $~$(1 \\cdot 3) : (4 \\cdot 1) \= 3 : 4$~$, meaning that the bottom pool has 3 parts of red water to 4 parts of blue water\.

The inverse of multiplication is division. To the mathematically steadfast this is completely obvious but I wager this is exactly the point where most non-mathematically inclined people will become confused and give up or will simply read on without absorbing the whole message. Maybe make this mathematical step more clearly?