# Lies, Lies

There are three prisoners in a prison. Prisoner 1 tells the truth $$20$$% of the time and lies the rest of the time, Prisoner 2 tells the truth or a lie with equal probability, and Prisoner 3 tells the truth $$\dfrac{9}{10}$$ of the time, and lies the rest. You decide to play a game. You arrange the prisoners in a line like so: $${\text{Prisoner 1}, \text{Prisoner 2}, \text{Prisoner 3}}$$. You go down the line, starting from Prisoner 1, and ask each of them a question. Whenever a prisoner tells the truth, you let them go, until only one remains. When the probability that Prisoner 1 is let go is expressed as a fraction in the form $$\dfrac{a}{b}$$ for coprime positive integers $$a$$ and $$b$$, what is the value of $$a+b$$?

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