Pan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm)


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1 STAT 225 FALL 2012 EXAM ONE NAME Your Section (circle one): Pan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm) Grant (3:30pm) Faye (4:30pm) Show your work on ALL questions. Unsupported work will NOT receive full credit. Decimal answers should be exact, or to exactly 4 decimal places. (Examples: if it is.25 use.25; if it is then use.5789.) You are responsible for upholding the Honor Code of Purdue University. protecting your work from other students. This includes Please write legibly. If a grader cannot read your writing, NO credit will be given. You are allowed the following aids: a onepage 8.5 x 11 handwritten (in your handwriting) cheat sheet, a scientific calculator, and pencils or pens. Instructors will not interpret questions for you. If you do have questions, wait until you have looked over the whole exam so that you can ask all of your questions at one time. You must show your student ID (upon request), turn in your cheat sheet and sign the class roster when you turn in your exam to your instructor. Turn off your cell phone before the exam begins. Question Points Possible Points Earned Cheat Sheet 1 Total 75
2 1. Suppose A, B, and C belong to the same sample space. Let A be an event that happens 20% of the time. Let B be an event that happens 30% of the time. Let C be an event that happens 45% of the time. Answer the following questions: (a) What is the smallest probability the intersection of A, B, and C can have? (b) What is the largest probability the intersection of A, B, and C can have? (c) Suppose A and B are mutually exclusive events. What is the smallest probability the union of A, B, and C can have?
3 2. An insurance company believes that people can be divided into three classes: highrisk, mediumrisk, and lowrisk. The company s statistics show that an highrisk person will have an accident at some time within a fixed 1year period with probability 0.5. However, this probability decreases to.3 for a person who is medium risk, and decreases further to 0.1 for a person who is lowrisk. Assume that 30% of the population is highrisk, 50% of the population is mediumrisk, and the remaining 20% of the population is lowrisk. (a) What is the probability that a new policyholder will have an accident within a year of purchasing a policy? (3 points) (b) What is the probability that the next 8 customers will not have an accident in their respective first years of coverage with this company? (3 points) (c) Suppose a customer has an accident within a year of purchasing her/his policy. What is the probability that (s)he is a highrisk customer? (4 points)
4 3. Suppose Mary has 10 different pairs of shoes (6 black and 4 brown), 4 different belts (2 black and 2 brown), 20 different tops, 6 different bottoms, and 5 different jackets. (a) How many different outfits can Mary make from her clothing collection (assuming she needs to wear at least one of each type of clothing)? (3 points) (b) Mary recently watched a fashion program that said you should only wear a brown belt with brown shoes. Assuming Mary follows this advice, how many different outfits can Mary make from her clothing collection now? (3 points) (c) Mary wants to go out on a date and cannot decide what to wear. Suppose there are 2 pairs of shoes (both black), 1 belt, 3 bottoms, 5 tops, and 2 jackets that are appropriate for her date. Since Mary can t decide what to wear, she asks her friend Jim to pick out an outfit from her entire clothing collection. Unfortunately, Jim has no fashion sense and doesn t know what articles of clothing are appropriate for a date. What is the probability that Jim picks out a dateappropriate outfit? (4 points)
5 4. In a certain dice game, the player rolls a fair die. If he gets a one, two, or three, he stops. If he gets any other number, the player gets a point and rolls one more time. If he gets a one or two on the second roll, he stops; any other number (3,4,5 or 6) and he adds one more point and his turn is over. (a) Create the pmf for the number of points the player can score during his turn (5 points) (b) Given that the player has scored at least 1 point, what is the probability that he scores 2 points? (3 points) (c) Suppose the player wins $5 if he scores 2 points, $2 dollars if he scores 1 point, and loses $2 dollars if he scores 0 points. How much money can the player expect to win in a single game? (3 points)
6 5. Choose the phrase from the following list that fits most accurately in the sentences below. Each phrase could be used once, more than once, or not at all. Your answer MUST be the letter that represents your chosen phrase. (2 points each) (A) combination (B) permutation (C) intersection (D) union (E) Law of Total Probability (F) InclusionExclusion formula (G) partition (H) independent (I) sample space (J) mutually exclusive (K) Bayes Theorem (L) conditional probability (a) If an event is said to be in both A and B, it is in the of A and B. (b) If P (A B) = P (A) + P (B), then A and B are. (c) Any unordered arrangement of r distinct objects from a collection of m objects forms a. (d) If P (B A) = P (B), then A and B are. (e) The set of all possible outcomes for a random experiment is called the.
7 6. (Adapted from ASW Chapter 4, problem 10) Many students accumulate debt by the time they graduate from college. Shown in the following table is the percentage of graduates with debt and the average amount of debt for these graduates at four universities and four liberal arts colleges U.S. New and World Report, America s Best Colleges, Suppose that each school has an equal number of graduates. University % with debt Amount($) College % with debt Amount($) Pace 72 32,980 Wartburg 83 28,758 Iowa State 69 32,130 Morehouse 94 27,000 Massachusetts 55 11,227 Wellesley 55 10,206 SUNY  Albany 64 11,856 Wofford 49 11,012 (a) If you randomly chose one of these eight institutions for a followup study on student loans, what is the probability that you will choose an institution with more than 60% of its graduates having debt? (1 point) (b) If you randomly chose a graduate with debt from one of these eight institutions, what is the probability the graduate went to a liberal arts college? (4 points) (c) If you randomly chose a graduate from one of the four colleges, what is the expected amount of debt the graduate will have? (3 points) (d) If you randomly chose a graduate from one of the four colleges, what is the variance in the amount of debt the graduate will have? (4 points)
8 7. A standard deck of poker playing cards has four suits (Hearts, Clubs, Spades, and Diamonds). Within each suit, there are 13 face values (2 through 10, Jack, Queen, King, and Ace), for a total of 52 cards. Suppose you are dealt 5 cards (without replacement) from this deck. (4 points each) (a) What is the probability that each of the 5 cards is different (in value)? (b) What is the probability that you were dealt a full house (3 of one face value, and 2 of another face value)? (c) What is the probability that you were dealt a Royal Flush (10, Jack, Queen, King, Ace of the same suit)?
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