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)$~$.