Alexander Bogolmony tweeted this problem:
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)
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
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.