CE 1030 Midterm Review Key

Transcription

1 1. a) C(10,3) = 120 b) C(15,3) = 455 c) CE 1030 Midterm Review Key 1 pt. Indication that combination is being used in part (a), Ex. C(10,3) or 10! or 10C3 or words indicating order is not important 1 pt. Correct answer for part a) = 120 = 10! 3! 7! points off for calculation error if correct work is shown 1 pt. Indication that combination is being used in part (b) 1 pt. Correct answer for part b) = 455 = 15! 3! 12! points off for calculation error if correct work is shown 3! 7! with formula or numbers. No with formula or numbers. No 1 pt. Correct answer for part c) or a ratio of answer to parts a) and b) even if the answers to part a) and b) were incorrect. 2. Rain 0.25 Fog b) The circles overlap. The overlap represents the probability that today is both rainy and foggy c) The probability of rain OR fog is 0.65 d) The probability of neither rain nor fog is pt. student drew a Venn diagram (doesn't have to be right) 0.5 pt. student labeled circles with Rain and Fog 1 pt. correct probabilities included in the table 0.5 pt. state or show in Venn diagram that circles overlap 0.5 pt. state that overlap represents days (or probability) with rain AND fog 1 pt. correct answer for (c) = 0.65 (no partial credit) 1 pt. correct answer for (d) = 0.35 (no partial credit) Page 1 of 8

2 3. b) Probability A did NOT write the exam is: Geometry = Prof B 0.20 No Geometry 0.90 Geometry 0.80 No Geometry 0.20 Prof A and Geom = 0.05 Prof A and no Geom = 0.45 Prof B and Geom = 0.16 Prof B and no Geom = 0.04 c) Probability there is geometry given that B or C wrote the exam is: = 0.62 d) Probability A or C wrote the exam given there is NO geometry is: = pt. for correctly labeled tree diagram Geometry 0.50 No Geometry 0.50 Prof C and Geom = 0.15 Prof C and no Geom = pt. correct answer for (b) = 0.5. No work needs to be shown. Answers of or are also sufficient for full credit. 1 pt. work for (c) indicating a conditional probability is required 0.5 pt. correct answer for (c) = 0.62 No work needs to be shown. Any correct, equivalent representation is accepted for full credit. 1 pt. work for (d) indicating a conditional probability is required 0.5 pt. correct answer for (d) = No deduction for incorrect rounding. Any correct, equivalent representation is accepted for full credit. Page 2 of 8

3 4. a) Many possible correct answers. For example: (3,2) then (1,4) then (1,1) b) Many possible correct answers. For example: (3,2) then (1,6) then (1,1) c) Probability of rolling a sum of five is 4 36 or 1 9 d) Probability of getting a sum of 5 exactly twice in three rolls is: 3 ( 4 36 ) ( 4 36 ) ( ) = pt. for correct answer for (a), no partial credit, must have results of 3 rolls, exactly 2 of which sum to 5. 1 pt. for correct answer for (b), no partial credit 0.5 pt. for indicating or using a correct probability of rolling a sum of 5 which is 4/36 or 1/9 or pt. correct answer of 4/36 or 1/9 1 pt. for correctly using product rule to find probability of two 5's and one non pt. for multiplying by 3, or adding 3 summands, indicating knowledge that there are 3 "ways" to roll two 5's and one non-5 5. a) correct b) incorrect. This situation would be modeled as y = 5.49x z because both the amount of pecans and the number of bags of sugar are unspecified. c) incorrect. This situation would be modeled as y = 2.50x because the number of bags of sugar are unspecified and he is buying one pound of pecans 1 pt. for circling correct on part a) 1 pt. circling incorrect on part b) 1 pt. for correct reason 1 pt. circling incorrect on part c) 1 pt. for correct reason 6. a) {Canoe, Car, Computer, Crayon} b) {Desk, Shoe, Strawberry, Tiger} c) {Bus, Ship} d) {Airplane, Boat, Camel, Canoe, Car, Computer, Crayon} 1 pt. uses {} 1 pt. correct answer to part a) 1 pt. correct answer to part b) 1 pt. correct answer to part c) 1 pt. correct answer to part d) Page 3 of 8

4 7. a) OR P(5,5) OR 5P 5 = 120 different possible orders he can order the ice cream scoops b) 5C 3 OR C(5,3) = 10 different ways he can combine three scoops of the five flavors of ice cream c) 1 10 = 0.10 There is a 10% chance Alex chooses chocolate, vanilla and mocha 1 pt. to realize part a) is a permutation 1 pt. correct answer part a) 1 pt. for knowing it s a combination (dividing out replicates) 1 pt. for correct answer part b) 1 pt. for correct answer part c) 8. Football M 9:00 W 10:00 W 11:00 Marching Band M 9:00 W 10:00 Color Guard W 11:00 F 12:00 F 12:00 Option 1. M 9:00 FB W 10:00 MB CG 2. M 9:00 FB W 10:00 MB W 11:00 CG 3. M 9:00 FB W 10:00 MB F 12:00 CG 4. W 11:00 FB M 9:00 MB CG 5. W 11:00 FB M 9:00 MB F 12:00 CG 6. W 11:00 FB W 10:00 MB CG 7. W 11:00 FB W 10:00 MB F 12:00 CG 8. F 12:00 FB M 9:00 MB CG 9. F 12:00 FB M 9:00 MB W 4:00 CG 10. F 12:00 FB W 10:00 MB CG 11. F 12:00 FB W 10:00 MB W 11:00 CG F 12:00 M 9:00 W 10:00 F 12:00 W 11:00 1 pt. student drew a tree diagram (doesn't have to be right, just has to be a tree) 1 pt. tree has 3 levels, one for each student group 2 pt. tree is correct - realizes that some branch options are not valid ex: can't schedule both Football and Marching Band Monday at 9:00 W 11:00 1 pt. correct list of 11 possible schedules (overlook "typos") Page 4 of 8

5 9. a) E = (0.1 6) + ( ) + (0.4 5) + (0.1 10) + (1 5) = 0.35 If you place a $5 bet, in the long run you should expect to lose $0.35 on each bet b) E = (0.1 12) + (0.3 3) + (0.4 10) + (0.1 20) + (1 10) = 0.70 Option B is not fair. You should expect to lose $0.70 on each bet. c) Option B is better for the carnival owner. He expected to take in more money from each bet with Option B 1 pt. correct answer for part (a) 1 pt. correct interpretation for part (a) 1 pt. correct calculation of expected value of option B 1 pt. correct explanation in part (b) 0.5 pt. circle Option B 0.5 pt. correct explanation in part (c) 10. a) y = m b) Budget is represented with the dashed line c) Slope represents the cost per mile you drive the truck d) Budget is less expensive if you plan to drive less than about 200 miles. The line representing the cost to rent from Budget is less than (or below) the line representing the cost to rent from Enterprise for the first 200 miles. 1 pt. correct equation in a) 0.5 pt. circle dashed 1 pt. correct explanation for b) 1 pt. slope is cost per mile 1 pt. Budget is less expensive when you drive less than 200 miles 0.5 pt. reasoning (referencing intercept or where lines cross) [FYI: The lines intersect at (200.05,168.02)] Page 5 of 8

6 AT&T 10 IBM GE pt. for Venn diagram with 3 overlapping circles 0.5 pt. for labeling circles as IBM, AT&T and GE 0.5 pt. for value of 5 in the 3-way intersection of the circles b) = 25 people did not invest in either IBM or GE c) 15 people invested in just IBM d) = 40 people invested in IBM or GE, but not AT&T 1 pt. for correct answer to (b) = 25 (no partial credit) 1 pt. for correct answer to (c) = 15 (no partial credit) 1 pt. for correct answer to (d) = 40 (no partial credit) 12. a) ( ) ( ) = = 0.84 million per year b) five years between 2009 and 2014, so the sales should decrease by another 4.2 million to 7.8 million OR 5 years 0.84million year = 7.8 million sales c) y = x = 4.2 million decrease d) Let y = 0. 0 = x x = 19.4 Sometime in 2023 sales of direct satellite receivers should be zero. 1 pt. correct answer for part (a) 1 pt. correct answer for part (b) 1 pt. correct equation for part (c) 1 pt. correct answer for part (d). Half credit if the student computes that in 19.4 years past start the sales will be 0 but does not translate that into the year pt. for correct answer and reasoning for part (e) e) No, it doesn t make sense for the sales to be a negative number, which would happen in 2024 if the change in sales is constant. Page 6 of 8

7 13. a) E = (25 125)+(30 150)+(45 175)+(15 200)+(10 225) = = $166/day b) 4550/26 = $175 c) $ days = $4316 day No, Josie should not expect to make enough during her first month to cover her living expenses. She needs to earn $175 per day and she s should only expect to earn $166 per day. Alternatively, she would expect to earn $4316 in the month and she needs $4550. d) 4550/166 = 27.4 days e) ( )/( ) = 70/125 = pt. correct answer for part (a) 1 pt. correct answer for part (b) 1 pt. answer no with reasoning comparing either per day or per month needs to expectations. 1 pt. correct answer for part (d). Answer may be given as 28 days or 27.4 days (or more decimals). Half credit if student answers 27 days. 1 pt. correct answer for part (e) 14. x: x is a number less than a) A = { } or equal to five b) B = {y: y is an even number less than or equal to 20} c) The cardinality of Set B is 10 and the cardinality of Set C is 4, so the cardinality of Set B is greater than the cardinality of Set C d) A B = {1,2,3,4,5,6,8,10,12,14,16,18,20} e) A C = {5} f) B = {1,3,5,7,9,11,13,15,17,19} 0.5 pt. correct set builder format for A or B including braces and x: 0.5 pt. correct description for A 0.5 pt. correct description for B 0.5 pt. circle greater than 0.5 pt. correct explanation for part c) 1 pt. correct set notation for parts d), e), f) 0.5 pt. correct elements in part d) 0.5 pt. correct elements in part e) 0.5 pt. correct elements in part f) Page 7 of 8

8 15. a) P(member of the crew) = b) P(survive) = c) P(survived given first class ticket) = d) P(third class ticket and survived) = e) First class passengers were much more likely to survive than third class passengers 1 pt. correct answer to part a) 1 pt. correct answer to part b) 1 pt. correct answer to part c) 1 pt. correct answer to part d) 1 pt. correct answer to part e) Page 8 of 8