STAT 225 Fall 2009 Exam 2 Your name: Your Instructor: Your class time (circle one): 7:30 8:30 9:30 10:30 11:30 12:30 1:30 2:30 3:30 4:30 Note: Show your work on all questions. Unsupported work will not receive full credit. All answers should be in decimal form and taken out to at least two decimal places. Do NOT leave answers in the form of combinations, factorials or exponentials. You are responsible for upholding the Honor Code of Purdue University. This includes protecting your work from other students. You are allowed the following aids: a one-page 8 ½ x 11 handwritten (in your handwriting) cheat sheet, a scientific calculator, and pencils. 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) and turn in your cheat sheet when you turn in your exam to your instructor. Turn off your cell phone before the exam begins! Question Points Possible Points Received 1 2 3 4 5 6 7 8 9 Cheat Sheet Total 100
1. You and a group of 5 friends decide to go out for pizza and bowling. On any given turn the probability a person will get a strike is 0.2 and the probability a person will get a spare is 0.6. On rare occasions, the chute will not return a ball with probability 0.005. Each turn is independent of any other. State the distribution and the parameters for each of the following situations. If an approximation can be made, to receive full credit you must name both the exact and approximate distributions. (3 points each) a. Upon arriving, the 6 of you go to a rack containing different colored bowling balls to make your selections. There are a total of 40 balls to choose from, 15 of which are black. You each select a bowling ball at random. Let X be the number of black bowling balls in your group. b. On your first turn let X be a success if you get a strike. c. You finish one game and had a total of 10 turns. Let X be the number of spares you had in the game. d. In the last hour there have been 3,000 turns played among all those in the Bowling Alley. Let X be the number of times a chute would not return a ball in those 3,000 turns. e. After your first two games, the owner allows you and your group to go in the back where there are a total of 400 bowling balls, of which 90 glow in the dark. The six of you each make a new selection of bowling balls. Let X be the number of bowling balls chosen that glow in the dark. f. You and each member of your group will eat an average of 2 slices of pizza. Let X be the number of slices of pizza that will be eaten.
2. Under every cap of a Snapple bottle, there is a little known fact however, only 30% of the facts are actually true. You purchase a 24 pack of Snapple and all Snapple bottles are independent of one another. Let T be the number of caps that contain a true fact. (3 points each) a. State the distribution and parameters of T. b. What is the probability that of the Snapple bottles you purchased, between 7 and 9 (inclusive) facts are actually true? c. What is the expected value and variance of number of facts that are actually true in your 24 pack of Snapple?
3. Given the PMF chart below, answer the following questions: (3 points each) y -2-1 0 4 7 P(Y=y) 0.15 0.35 0.25 0.2 0.05 a. What is the probability Y is not negative? b. What is the probability Y is between -1.2 and 3.8? c. What is the expected value of Y? d. What is the standard deviation of Y? e. W = Y (absolute value of Y). Create a PMF chart of W.
4. At the Baltimore aquarium, there are a total of 40 penguins, 15 of which are King Penguins. Every day at 2pm, a group of 5 penguins is selected to perform at the Penguin Show. Let K be the number of King Penguins that are selected. a. State the distribution and parameters of K. (3 points) b. What is the probability that on a given day at least 2 King Penguins are selected for the Penguin Show? (3 points) c. If this process is repeated for one week, what is the probability that exactly 5 of the 7 days have at least 2 King Penguins in the Penguin Show? State the distribution and parameters you are using. (4 points) 5. For a random variable X, let E(X) = 4 and Var(X) = 9. (3 points each) a. Find E(3X 5) b. Find Var(7 2X) c. Find E(X 2 )
6. A set of old Christmas lights has 500 individual lights, and unfortunately, the entire strand of lights will not work unless ALL lights are working. The probability that an individual light is not working is 0.003 and all lights will work independently of one another. (3 points each) a. What is the approximate probability that exactly 1 light is not working? State the distribution and its parameters you are using and why you can use it. b. What is the approximate probability that the strand of lights is not working? c. Given that the strand is not working, what is the approximate probability that exactly 1 light is not working?
7. You roll an 8-sided blue die, and a 4-sided orange die. Find the following probabilities: (it may help to draw a grid) (3 points each) a. P(blue die is not a multiple of 3) 6/8 b. P(blue die is a 3 and orange die is a 4) 1/8*1/4 = 1/32 10/32 c. P(sum is 4 or the orange die is a 2) d. P(sum is at least 10 blue die is 8) 3/4
8. In a certain company, 75% of employees are male 42% of employees are CEO s 15% of employees are female and are not CEO s a. What is the probability an employee is female given the employee is a CEO? (Drawing a Venn Diagram may help, but is not required). (4 points) P(F CEO) = 10/42 =.24 b. Are the events that an employee is male and that an employee is a CEO independent? Use formulas to support your answer and earn full credit. (4 points) No, P(F) =.25 not equal to P(F CEO) =.24
9. The three leading credit card companies are Visa, MasterCard and Discover. People using any of these credit cards were asked to pick which was their primary credit card. A person is twice as likely to use Visa as their primary credit card than Discover. A person is equally likely to use Visa as their primary credit card as MasterCard. 20% of those with Visa as their primary credit card, missed their last monthly payment 10% of those with MasterCard as their primary credit card, missed their last monthly payment 5% of those with Discover as their primary credit card, missed their last monthly payment a. What is the probability a person selected at random uses either Visa or Discover as their primary credit card? (3 points) P(V)=P(M) =.4 P(D) =.2 P(V or D) =.4 +.2 =.6 b. What is the probability that a person selected at random missed their last monthly payment? (drawing a tree diagram may help but is not required) (3 points) P(Miss) = P(Visa and Miss) + P(MC and Miss) + P(Disc and Miss) =.4*.2+.4*.1+.2*.05=.13 c. Given that a person missed their last monthly payment, what is the probability they did not use Discover as their primary credit card? (3 points) P(Visa or MC Miss) =.4.2+.4.1.13 =.92