Exercise 1: Condorcet's Jury Theorem (30 points) A person has allegedly committed a crime and a trial will take place for a jury of 3 to declare him either innocent or guilty. Previous to the trial start, none of the jury members has any knowledge of the situation and being impartial jurors, if they had to convict now, they would decide with the following probabilities: $$ P_{j}(G \mid \text { guilty })=P_{j} (I \mid \text { innocent })=0.5 $$ where $js refers to the jury number, $G$ is a guilty vote $I$ is an innocent vote), and both probabilities are conditional probabilities over the underlying truth. (a) (10 pts) What's the probability the jury would make the correct choice if they had to decide at this moment? Library Course Page S.P.PB. 342