choose a fair coin, then get three heads and one tail tossing

the fair coin; and second, choose an unfair coin, then get

three heads and one tail tossing the unfair coin:

P(3H,T)=P(F)*P(3H,T|F)+P(U)*P(3H,T|U)= (8/10)(4/2^4)+

(2/10)(4*0.9^3*0.1) =0.25832

P(F|3H,T)= P(F)*P(3H,T|F) / P(3H,T)= (8/10)(4/2^4/025832=

0.7742

heads and two tails:

P(2H,2T)=P(F)*P(2H,2T|F)+P(U)*P(2H,2T|U)=(8/10)(6/2^4)+

(2/10)(4*0.9^2*0.1^2) =0.30648

Then, Bayes’ formula immediately gives us the answer:

P(F|2H,2T)= P(F)*P(2H,2T|F) / P(2H,2T)=(8/10)(6/2^4)/

0.30648≈0.6525