Probability Question

lmao AoPS homework

P("choose fair coin") = 24/25

P("choose unfair coin") = 1/25

P("6H under the condition fair coin")$= 2^{-6} = \frac{1}{64}$

P("6H under the condition unfair coin")=1

let a=P("choose fair coin") * P("6H under the condition fair coin") = 24/25 * 1/64

let b=P("choose unfair coin") * P("6H under the condition unfair coin") = 1/25

$\frac{b}{a+b} = \frac{m}{n} = \frac{8}{11}$

m+n=19