A coin is either fair (0.5 probability of heads) or unfair (0.6 probability of heads).
Fair and unfair coins are identical at sight.
If I throw it and get 3 heads out of 3 throws, which is the probabilty that it is unfair?
The total probability (1) of my event that I observed (3 heads) is
(probability of it being fair)*(probability of my outcome if it is fair) + (probability of it being UNfair)*(probability of my outcome if it is UNfair) = 1 probability of my outcome if it is fair = (1/2)^3 = 1/8 probability of my outcome if it is UNfair (6/10)^3 = (27/125) probability it is fair = F probability it is UNfair = 1 - F
$F(1/8) + (1-F)*(27/125) = 1$
But F is bigger than one and negative (F = -112/13) !
Please help me to understand the fault in my reasoning.