So this is part of a larger question on this game theory problem set, but I'm really just having troubles with this one part—apologies, it's probably really elementary. We have a defendant on trial in front of four judges. He is either guilty or innocent. Each judge receives a "signal" about whether he is guilty or innocent, and that signal is right with probability 0.9, ie. if he is guilty then each judge has a 0.9 chance of receiving a "guilty signal" and a 0.1 chance of receiving a "not guilty" signal. I need to find the probability that, given that the defendant is not guilty, exactly three judges receive the guilty signal.
So at first I thought it was easy enough: $0.1\times 0.1\times 0.1 \times 0.9$, but then I realized that this is only the probability that, if we were to order them Judge 1, 2, 3, 4, Judges 1, 2, and 3 receive the guilty signal and Judge 4 receives the not guilty signal. Is that correct? So, my question is, do I need to somehow incorporate all the possible orderings of the judges into this calculation?
Thanks for your help, and again, sorry if this is very easy.