Predicting heads after a coin passes two flips

A hat holds three coins: one shows heads with probability $\tfrac13$, one with probability $\tfrac23$, and one has heads on both sides. A coin is picked at random and tossed twice, giving heads both times. That same coin is now tossed 12 further times. What is the expected number of heads in those 12 tosses?

$72/7$. Bayes with likelihoods $\tfrac19,\tfrac49,1$ gives posterior weights $\tfrac1{14},\tfrac4{14},\tfrac9{14}$. The posterior mean heads probability is $\tfrac{1}{14}\left(\tfrac13+\tfrac83+9\right)=\tfrac67$, so 12 tosses yield $\tfrac{72}{7}$ heads on average.