calculate the Probability: What is the probability that no one makes the right decision?

Question

I'm trying to solve an exercise from the stochastic project. Is anybody already done such exercises?

Consider a jury trial in which it takes 9 of the 12 jurors to convict the defendant; that is, in order for the defendant to be convicted, at least 8 of the jurors must vote him guilty. If we assume that jurors act independently and that, whether or not the defendant is guilty, each makes the right decision with probability θ (here could be 0.75 or 0.80), what is the probability that the jury renders a correct decision? the questions are: 1) How many jurors making the right decision can be expected? 2) What is the probability that no one makes the right decision?

Thanks! BR Tony

Answer 1

1. How many jurors making the right decision can be expected? It's 12*θ
2. What is the probability that no one makes the right decision? It's (1-θ)^12