Consider a production system consisting of three bernoulli machines. Each machine is up during a cycle time with probability 0.9 and down with probability .1. Machines are up or down independently from each other.
(a) What is the probability that all three machines are up simultaneously given that at least two machines are up?
(b) What is this probability, given that at least one machine is up?
(c) Explain the reason for the difference between these two numbers.