We need to first calculate the probabilities for the unfair coin:  

P(H) = 5∙P(T)
P(H) + P(T)=1
P(T) = 1-P(H)
P(T) = 1-5∙P(T)
6P(T)= 1
P(T) = 1/6
P(H) = 5∙(1/6)
P(H) = 5/6

There are two possibilities

1. He selects a fair coin and it comes up tails

P(fair coin) = 2/3
P(tails on fair coin) = 1/2

P[(selects fair AND tails on fair coin) OR (selects unfair AND tails on unfair coin)]

AND means we multiply and OR means we add.

(2/3)(1/2) + (1/3)(1/6) 

That calculates to be 7/18 

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P(fair|tail) = P(fair AND tail)/P(tail) = [(2/3)(1/2)]/(7/18) = 6/7

Edwin