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).