Professor's misery

Parinya needs to go to a trial for teaching too much mathematical content in the CS department. In this trial, there are $k$ members of the jury (where $k \geq 10$). Parinya would be found innocent if at least one jury member finds him innocent; otherwise, he would be found guilty.

The jury's decisions are (possibly coordinated) random processes, i.e. a member's decision may or may not depend on the other's decision (so this means, we don't know the probability space!). Let $A_i$ be event that jury-$i$ finds Parinya innocent. Parinya has no knowledge about the jury's random process except for the fact that ${\mathbb P}[A_i] = 1/k$ (i.e. each jury's member finds him innocent with probability $1/k$). Parinya is trying to estimate his chance and decide whether he should flee to Belize instead of going through a trial that he would lose.

Exercise 114

Calculate the expected number of jury's members that would find him innocent.

Exercise 115

If each jury makes their decision independently, what is the probability that Parinya would be found guilty? Formally define this probability space.

Exercise 116

Describe a scenario (a probability space) where the jury's random process could find Parinya guilty with probability as close to one as possible. Argue formally that your scenario is really the worst possible for Parinya¹.

Footnotes

1. Keep in mind that your random process must be consistent with the fact that ${\mathbb P}[A_i] = 1/k$ ↩