• 1. Using the definitions and properties discussed in class, prove the following statements. Note, it is not sufficient to simply plug in numbers and show that the statements hold, you need to show that the statements algebraically follow the definitions/formulas given in class. (a) For events A and B, show that P(A ∩ B) ≥ P(A) + P(B) - 1. (b) For events A and B, show that p(B)/p(A) = p(B | A)/p(A | B)
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• 2. A certain state has license plates showing three numbers before three letters (numbers and letters can both be repeated). (a) How many different license plates are possible? (b) If you request a plate in random, what is the probability you get 000AAA? (c) Is the probability you just calculated larger or smaller than that of any other plates?
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• 3. A portfolio of five stocks is to be created from a group of ten technology stocks, five pharmaceutical stocks, and twenty retail stocks. If the portfolio was formed randomly, then what is the probability that it contains two technology stocks, two pharmaceutical stocks, and one retail stock?
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• 4. Suppose that the probability of FDA approval for a certain drug (drug A) is 0.15 and that the probability of FDA approval for another drug (drug B) is 0.35. If the probability of approval for either drug A or drug B (or both drugs) is 0.4, then what is the probability of drug A being approved given that drug B was approved?
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• 5. In this problem we consider rolling two 5-sided dice and taking the sum of the values. To generate the situation, you can run the following R code: S _- expand.grid(die1=1:5, die2=1:5) S _- transform(S, sum=rowSums(S)) Consider the value in the last column (the sum of the two dice). (a) What is the sample spaces? (b) What are the probabilities for each value in the sample space? (c) What is the probability of rolling a five given that the first die was an odd number?
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• 6. In a certain industry, 60 percent of all labor-management disputes are over wages, 15 percent are over working conditions, and 25 percent are over fringe issues. 47.5 percent of all disputes are resolved without a strike. In addition, 45 percent of the disputes over wages are resolved without strikes, 70 percent of the disputes over working conditions are resolved without strikes, and 40 percent of the disputes over fringe issues are resolved without strikes. What is the probability that, if a labor-management dispute in this industry is resolved without a strike, it was over wages (Use Bayes' Rule)?