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Question 1 A delegation of three is to be chosen from the untenured faculty of the MIT Economics Department (numbering ten) to represent the department in an Institute-wide committee. In how many ways a) can the delegation be chosen? b) can it be chosen, if two people refuse to go together? c) can it be chosen, if two particular members insist on either both going or neither going? d) can it be chosen, if two people must be chosen from MIT assistant faculty (6 professors) and one person must be chosen from visiting assistant faculty (4 professors)? Question 2 In the seventeenth century, Italian gamblers used to bet on the total number of spots rolled with three dice. They believed that the chance of rolling a total of 9 ought to equal the chance of rolling a total of 10. They noted that altogether there are six combinations to make 9: (1,2,6), (1,3,5). (1,4,4), (2,3,4), (2,2,5), and (3,3,3). Similarly, there are six combinations for 10: (1,4,5), (1,3,6), (2,2,6), (2,3,5). (2,4,4), (3,3,4). Thus, argued the gamblers, 9 and 10 should have the same chance. Empirically, they found this not to be true, however. Galileo solved the gambers' problem. How? a) How many permutations of three dice are there that sum to 9? b) How many permutations of three dice are there that sum to 10? c) How many total permutations of three dice are there? What was Galileo's solution? Explain. Question 3 Venn diagrams or set diagrams are diagrams that show all hypothetically possi- ble logical relations between a finite collection of sets (groups of things). Venn diagrams were invented around 1880 by John Venn. They are used in many fields, including set theory, probability, logic, statistics, and computer science (Wikipedia: http://en.wikipedia.org/wiki/Venn_Diagram)- a) Draw a Venn diagram for the three events A, B, and C contained in the sample space S and properly label all possible union and intersections of events. b) Draw a Venn diagram for the three events A, B, and C contained in the sample space S and properly label all possible combination of events where AnBAC =0. c) Try (but don't spend too much time-it's just for fun) to draw a complete Venn diagram for the four events A, B, C, and D contained in the sample space S where you include all possible unions and intersections of events. How many mutually exclusive regions should such a diagram include? d) How many mutually exclusive regions should such a diagram with k E N events include? Question 4 Does a monkey have a better chance of rearranging "ACCLLUUS" to spell "CALCULUS," or of rearranging "AABEGLR" to spell "ALGEBRA?" (2 points.) Question 5 In Lecture 1, you learned about event partitions. Give three different examples of partitions of a single draw from a deck of playing cards. Question 6 The MIT football team plays 12 games in a season. In each game they have } probability of winning, : probability of losing, and ; probability of tying. Games are independent. What is the probability that the team has 8-3-1 record? (6 wins, 4 losses, and 2 ties)Question 7 You and your friends just rented a car from Enterprise for an 8,000 mile cross- country road trip to see all of the sights from from Boston Harbor to the Golden Gate Bridge. Your rental car may be of three different types: brand new (and not a lemon), nearly 1 year old, or a lemon (bound to break down). That many miles can be demanding on a rental car. If the car you receive is brand new 2 (New), it will break down with probability 0.05. If it is one year old (One), it will break down with probability 0.1. If it is just a lemon (Lemon), it will break down with probability 0.9. The probability that the car Enterprise gives you a car that is New, One, or Lemon is 0.8, 0.1, and 0.1, respectively. Compute the probability that your car is going to break down on your road trip. Question 8 a) Bayes' formula is really important. Write down Bayes' formula and de- scribe it in words. Further, here are a couple of common applications. b) Suppose that five percent of men and 0.25 percent of women are color blind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if there were twice as many males as females? c) Suppose that there exists an imperfect test for Tuberculosis (TB). If some- one has TB, there is a ninety-five percent chance that the test will come up "red." If someone does not have TB, there is only a two percent chance that the test will come up red. Finally, the chance that anyone has TB is, say, five percent (in the United States; in other countries Tuberculosis is endemic). Once someone takes the test and it comes up red, what is the probability that they have TB?Question 1 Recall that a random variable X has the binomial distribution if P(X = x) = (")D(1-p)"-=, where n is the number of trials and p is the chance of success. For the following crudely-disguised questions about the binomial distribution, do the following: Define p, n, what the specific "trials" are, what "success" is. Then write down the relevant distribution and answer the specific question. 1. If 25 percent of the balls in a certain box are red, and if 15 balls are selected from the box at random, with replacement, what is the probability that more than four red balls will be obtained? 2. (3 pts.) Suppose an economist is organizing a survey of American minimum wage workers, and is interested in understanding how many workers that earn the minimum wage are teenagers. Suppose further that one out of every four minimum wage workers is a teenager. If the economist finds 80 minimum wage workers for his survey, what's the probability that he interviews exactly 14 teenagers? 35 teenagers? What's the probability that he gets at least 5 teenagers in his survey? 3. (Bonus Question) A city has 5000 children, including 800 who have not been vaccinated for measles. Sixty-five of the city's children are enrolled in a day care center. Suppose the municipal health department sends a doctor and nurse to the day care center to immunize any child who has not already been vaccinated. Find a formula for the probability that exactly & of the children at the day care center have not been vaccinated. (Hint: This is not exactly a binomial distribution problem.) 1In the debate over the minimum wage, one point that is always made is that a lot of workers on the minimum wage are middle-class teenagers. Since most of them are not supporting a family, the harms of an increase in the minimum wage outweigh the benefits. This point never convinces anyone. Question Two Suppose you flip a weighted coin (probability of heads is p and probability of tails is q =1 -p) n times. 1. What is the probability that you get a particular ordering of k heads and n - k tails? 2. What is the probability that you get k heads and n - k tails? 3. Let X =the number of heads in n flips. What is the probability density function of X? 4. What does this question have to do with the MIT Beaver's football team ques- tion from Problem Set #1? Explain. Suppose now that you have a hat with two coins, one weighted as above and one fair. You choose one at random and flip that one n times. 1. Let Y =the number of heads in n flips. What is the PDF of Y? 2. What is the probability that you chose the fair coin given that Y = k?Question Three Suppose that two balanced dice are rolled. Determine and sketch the probability distribution of each of the following random variables. 1. Let X denote the absolute value of the difference between the two numbers that appear. 2. Let Y denote the product of the two numbers that appear. 3. Let Z denote the number of even numbers that appear. Question Four Suppose that you have just purchased a new battery for your smoke detector, and the life of the battery is a random variable X, with pdf fx(x) = ke-=/B where r E (0, co). Assume that t and s are real non-negative numbers. 1. Use the properties of a pdf to find the value of k. 2. Find an expression for P(X > t). 2 3. Find an expression for P(X 2 t + s|X 2 8). 4. Suppose that your batteries have lasted s weeks without dying. Based on your above answers, are you more concerned that the battery is about to die that you were when you first put it in? Question Five Suppose we investigate the pattern of genetic inheritance for the color of an exotic flower which has either blue or red blossoms. Since the flower lives in a close symbiotic relationship with the very shy squirrel monkey which can't be held in captivity, there is no way of doing a controlled laboratory experiment to answer the research questions. Each flower carries the color genes of both its "father" and its "mother," so its genetic information can be described as a pair of genes (GM, GF) as given by the following table: "Father" B R "Mother" B (B,B) (B,R) R (R,B) (R,R) The phenotype corresponding to red blossoms, R, is said to be dominant if any flower which contains at least one gene of the R type (e.g. the combination (R,B)) has red blossoms. Either the blue or the red phenotype is dominant, but before having seen a single specimen of the flower, we think that each possibility is equally likely. 1. Suppose we know beforehand that the R and the B alleles are equally frequent, i.e. Pi(B) = Pi(R) = , for i = F, M, and independent across the "parents," i.e. PFM(GF, GM) = PF(GF)PM(GM). If the blue phenotype B is dominant, what is the probability that a given specimen of the flower has red blossoms? What is the probability of red blossoms if the R is dominant? 2. It takes a lot of effort to find a single specimen of the plant, so all a well-funded two-month expedition by a team of MIT botanists to the Amazon could gather was a sample of 15 flowers. If R is dominant, what is the probability of 9 out of the 15 flowers having red blossoms? 3. Our expedition did in fact return with a sample of 9 red and 6 blue blossoms. Given that, what is the likelihood that the red phenotype is dominant? 4. At the same time, there is a lonely graduate student working in the same area for two entire years on the same research question. The graduate student is totally cut off from the outside world and doesn't know about the other expedition's findings yet, but bases his inference solely on his own sample. What is his posterior probability of R being dominant given that he found N flowers, out of which z have red blossoms? Show that this probability does not depend directly on N, but only the difference between the number r of red blossoms, and the number N - r of blue blossoms