Probability question please help me solve it and write on paper

2) (18 pts) A flight operator monitors the sky during his eight hour night shift. The targets are known to be Poisson distributed and on average there are two targets in four hours. a) What is the PMF describing the number of targets within his night shift? b) What is the probability there is at least one target within his night shift? 0) Assume the radar operator sees two targets and then falls asleep through the rest of his night shift. What is the probability he misses a target that night? 3) (33 points) An integrated circuit factory has three machines X, Y, and Z. The integrated circuit produced by each machine is tested and either a circuit is acceptable (a) or it fails (f). An observation is a sequence of three test results corresponding to the circuits from machines X, Y, and Z, respectively. For example, aaf is the observation that the circuits from X and Y pass the test and the circuit from Z fails the test. (a) What are the elements of the sample space of this experiment? (b) What are the elements of the sets XA = {circuit from X is acceptable} and YF = {circuit from Y fails}? (c) Are XA and YF mutually exclusive? Why or why not? ((1) Are XA and YF collectively exhaustive? Why or why not? (e) What are the elements of the sets C = {at least one circuit is acceptable} and D = {none of the circuits pass}? (i) Do C and D form a partition? Why or why not? (g) Assume the probability of an acceptable or a failing integrated circuit are equal, furthermore they are the same for all three machines. What are the corresponding probabilities P(XA), P(YF), P(C) and P(D)? 5) (15 points) Machines X,Y and Z produce 20%, 40% and 40% of the integrated circuits in a plant. They know that, respectively 20%, 5% and 10% of the integrated circuits produced by X, Y and Z are defective. (a) Draw the tree diagram for this problem. (b) What is the probability that a produced integrated circuit is found defective? (6) They pick an integrated circuit and see that it is defective. What is the probability it was produced by machine Z