If there is no reason to assume in advance that either coin is more likely to be the coin tossed once (i.e. the first coin) then all the (correct) solutions show that the first coin is more likely to be biased with a probability of 9/17 (=0.52941). Here is an explicit Bayesian network solution for the problem:
The above figure shows the result after entering the 'evidence' (i.e. one Head on the coin tossed once and two Heads on the coin tossed three times). The tables displayed are the conditional probability tables defined for the associated with the variables.
This model took just a couple of minutes to build in AgenaRisk and requires absolutely no manual calculations as the Binomial distribution is one of many functions pre-defined. The model (which can be run in the free version of AgenaRisk is here). The nice thing about this solution compared to the others is that it is much more easily extendible. It also shows the reasoning very clearly.
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