"I believe that this should be $(2 : 3 : 1)$ rat..."

We have three hypotheses, which we'll call $H\_{fair},$ $H\_{heads}$, and $H\_{tails}$ and respectively, with relative odds of $(1/2 : 1/3 : 1/6).$ The relative likelihoods that these three hypotheses assign to a coin landing heads is $(3 : 2 : 1)$; the relative likelihoods that they assign to a coin landing tails is $(2 : 1 : 3).$ Thus, the posterior odds for all three hypotheses are:

I believe that this should be $(2 : 3 : 1)$ rather than $(3 : 2 : 1)$.