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Math 1031 Sample Midterm 4A This is the format and directions that you will see on the cover page of the actual exam. Name: Discussion Section: Discussion Instructor: You may use a scientific calculator, but you may not use books, notes, graphing calculators, or your neighbors’ papers. Sign your name below to certify that you followed these instructions. Signature: Do all your work in the space provided on these sheets. If you need additional paper, attach it to these sheets. On the multiple choice questions, clearly indicate the answer that you choose. If your selection is not clear, you will not earn any points for that problem. Partial credit will be rewarded on the short answer problems. You will not earn credit for illogical, incorrect, or unsupported work, even if you miraculously arrive at the correct answer. If you are not certain how to do a problem, give it your best attempt so that you may earn some credit for moving in the right direction. Circle your final answer on the short answer problems. The exam will be graded out of 100 points. The point value for each problem is listed beside the problem number. There are 7 pages and 12 problems on the exam. Here are some formulas that you might find useful. P (E) = n(E) n(S) P (E ∩ F ) = P (E) · P (F ) (independence) P (E ∪ F ) = P (E) + P (F ) − P (E ∩ F ) P (E 0 ) = 1 − P (E) P (x successes) = C(n, x)px (1 − p)n−x (binomial probability experiment) ) P (E|F ) = P P(E∩F or P (E ∩ F ) = P (F ) · P (E|F ) (conditional probability) (F ) Ev = x1 p1 + x2 p2 + . . . + xn pn (expected value) 1 1. (10 points) In a group of 15 students, 9 are taking a chemistry class, 7 are taking a physics class, and 3 are in both chemistry and physics. If a student is selected at random from the group, what is the probability that the student is enrolled in chemistry or physics? Solution: 13 15 2. (10 points) A labrador retriever dog has puppies that are either yellow or black. For a certain labrador retriever, the probability of a puppy being yellow is 60%. If this labrador retriever has a litter of 8 puppies, what is the probability that 3 are black? Express your answer as a percentage, and round to the nearest percent. Solution: 28% 2 3. (12 points) (a) How many ways are there to select three letters from the letters A, B, C, D, E, and F? Give both the solution and the formula you used to find it. Solution: C(6, 3) = 20 (b) How many ways are there to break the letters A, B, C, D, E, and F into two three-element sets? Give both the solution and the formula you used to find it. Solution: C(6,3) 2 = 10 (c) How many ways are there to select three letters from the letters A, B, C, D, E, and F, if one of the letters selected is A? Give both the solution and the formula you used to find it. Solution: C(5, 2) = 10 4. (10 points) The value of a computer is $500. A one-year warrenty costs $75. You estimate that there is a 10% probability of having to replace the computer in the first year. What is the expected value of the warrenty? Solution: −$25.00 3 5. (10 points) 13 people are employed at a hospital. 4 of them are doctors and the other 9 are nurses. A committee of 5 people is selected at random from the 13 hospital employees. (a) How many such committess are there? Give both the solution and the formula you used to find it. Solution: C(13, 5) = 1287 (b) What is the probability that the committee includes at least one doctor? Express you solution as a percentage, and round to the nearest percent. Solution: 90% 6. (12 points) In a bakery, half the options on the menu contain nuts and 15% contain chocolate. If an item is chosen at random from the menu, the probability that it contains both nuts and chocolate is 7.5%. (a) Let E be the event that an item ordered from the menu contains nuts, and let F be the event that an item contains chocolate. Are the events E and F independent or dependent? Support your answer. Solution: independent, since P (E ∩ F ) = P (E) · P (F ) (b) Suppose you visit the bakery 10 times, and each time you randomly select an item from the menu. What is the probability that at least 2 of the items you order contain chocolate? Express you answer as a percentage, and round to the nearest percent. Solution: 46% 4 7. (6 points) Suppose that the probability of success in an experiment is 63%. If the experiment is repeated 15 times, what is the approximate probability of having 5 successes? (a) 20.51% (b) 2.93% (c) 1.72% (d) 1.43% Solution: d 8. (6 points) A 3-person committee is selected from the eight people Jan, Don, Hui, Pat, Kit, Bob, Sam, and Ida. How many ways are there to form a committee that includes Jan or Don, but not both? (a) 15 (b) 30 (c) 56 (d) 225 Solution: b 5 9. (6 points) A survey was given to drivers in North Dakota and South Dakota asking them whether they had ever hit a deer. The results are given in the following table. North Dakota South Dakota total hit a deer 18 26 44 never hit a deer 25 31 56 total 43 57 100 Based on the information in the table, what is the probability that a person in the survey has hit a deer, given that the person is from North Dakota? 9 50 9 (b) 22 (c) 18 43 69 (d) 100 (a) Solution: c 10. (6 points) There are six cars driving in a line. The cars are black, blue, green, red, tan, and white. How many orders could the cars be driving in if the red and white cars are neither first nor last? (a) 24 (b) 48 (c) 288 (d) 720 Solution: c 6 11. (6 points) Toss 2 dice. What is the probability of getting a sum less than 5? 1 12 (b) 19 (c) 16 (d) 29 (a) Solution: c 12. (6 points) How many permutations can be made from 3 identical A’s and 4 identical B’s? (a) 35 (b) 420 (c) 2520 (d) 5040 Solution: a 7