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Fundamentals. In each step, explain clearly what property or axiom you are using.(a) Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B).(b) Provethe“unionbound,”thatP(A1∪A2)≤P(A1)+P(A2).Underwhatconditionsdoesthe equality hold?(c) Provethat,forA1 andA2 disjoint,P(A1∪A2|B)=P(A1|B)+P(A2|B).(d) A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.You roll a fair six-sided dice twice and record the results, in order. (a) What is the sample space for this experiment?(b) What is the probability that the total (sum) of the results is equal to 10?(c) Given that the total is equal to 10, what is the probability that the first roll was a 4?(d) Given that the first roll was not a 4, what is the probability that the total is equal to 10?A device consists of two redundant subunits; only one needs to be working for the device to function. The first subunit has two parts W1 and W2, and the second has two parts W3 and W4. In each subunit, both parts must function for the subunit to work. The four parts are identical and equally reliable, with probability p = 0.9 of functioning, independent of each other.(a) What is the overall probability the device functions correctly?(b) Suppose you have available two ultra-reliable parts, with p = 0.99 that each functions. Which two parts in the device should you replace with these in order to maximize device reliability? Justify your answer by calculating the resulting reliability and explain your choice.At a factory, three robots (A, B and C) build devices. 20% of the devices made by A are defective, 10% of the devices made by B are defective, and 5% of the devices made by machine C are defective. Each robot builds the same number of devices per day. Let F indicate a defective device.(a) What is the probability P(F) that a given device produced in this factory is defective?(b) Given that a device is defective, what is the probability that it was produced by robot A?(c) Given that a device is not defective, what is the probability that it was produced by robot C?Generalize the Bayesian analysis of medical screening given in class. Let a rare disease D occur with probability P(D) = δ, with δ small. The test for this disease is accurate such that the probability of a correct result is 1 − ε, with ε small. Derive the probabilities of a true positive P(D|+) and a false positive, P(Dc|+). For this screening to be useful at a given δ, what is required of ε ?