CE 1030 Midterm Review-KEY.pdf

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Page 1 of 8

CE MATH 1030

Midterm Review-KEY

Rubric

1 pt. Indication that combination is being used in part (a), Ex. 𝐶(10,3) or 10! 3!∙7! or 10𝐶3 or words indicating order is not important

1 pt. Correct answer for part a) = 120 = 10!

3!∙7! with formula or numbers. No 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! with formula or numbers. No points off for calculation error if correct work is shown

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.

Rubric

0.5 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)

Rain

Fog

CE 1030 Midterm Review Key

1.

2.

a)

𝐶(10,3) = 120

b) 𝐶(15,3) = 455

c) 120₄₅₅≈ 0.264

0.25

0.15

0.25

0.35

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

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CE MATH 1030

Midterm Review-KEY

Rubric 1 pt. for correctly labeled tree diagram

1 pt. correct answer for (b) = 0.5. No work needs to be shown. Answers of 1-0.5 or 0.3+0.2 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) = 0.9375. No deduction for incorrect rounding. Any correct, equivalent representation is accepted for full credit.

3.

_{b) Probability A did NOT write the exam is:}

0.20 + 0.30 = 0.50

c) Probability there is geometry given that B or C wrote the exam is: 0.16+0.15_0.2+0.3

= 0.62

d) Probability A or C wrote the exam given

there is NO geometry is:

0.45+0.15

0.45+0.04+0.15

= 0.9375

Geometry 0.10 No Geometry

0.90

0.20

0.50 Prof B

0.20

Prof A and Geom

= 0.05

Prof A and no Geom

= 0.45

Prof B and Geom

= 0.16

Prof C and Geom

= 0.15

Prof B and no Geom

= 0.04

Prof C and no Geom

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CE MATH 1030

Midterm Review-KEY

Rubric

1 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 0.111

1 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-5

0.5 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

Rubric 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

Rubric 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) 4.

5.

6.

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 ₃₆4 or 1₉ d) Probability of getting a sum of 5 exactly twice in three rolls is: 3 (₃₆4) (₃₆4) (1 −₃₆4) = 0.0329

a) {Canoe, Car, Computer, Crayon}

b) {Desk, Shoe, Strawberry, Tiger}

c) {Bus, Ship}

d) {Airplane, Boat, Camel, Canoe, Car, Computer, Crayon}

a) correct

b) incorrect. This situation would be modeled as 𝑦 = 5.49𝑥 + 2.50𝑧 because both the amount of pecans and the number of bags of sugar are unspecified.

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CE MATH 1030

Midterm Review-KEY

Rubric 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)

Rubric

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

1 pt. correct list of 11 possible schedules (overlook "typos")

7.

8.

Option

1. M 9:00 FB W 10:00 MB M 10:00 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 M 10:00 CG 5. W 11:00 FB M 9:00 MB F 12:00 CG 6. W 11:00 FB W 10:00 MB M 10:00 CG 7. W 11:00 FB W 10:00 MB F 12:00 CG 8. F 12:00 FB M 9:00 MB M 10:00 CG 9. F 12:00 FB M 9:00 MB W 4:00 CG 10. F 12:00 FB W 10:00 MB M 10:00 CG 11. F 12:00 FB W 10:00 MB W 11:00 CG a) 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 OR 𝑃(5,5) OR 5P5 =

120 different possible orders he can order the ice cream scoops

b) 5C3 OR 𝐶(5,3) = 10 different ways he

can combine three scoops of the five flavors of ice cream

c) ₁₀1 = 0.10

There is a 10% chance Alex chooses chocolate, vanilla and mocha

M 9:00 W 10:00

M 10:00

F 12:00

M 10:00

F 12:00

M 10:00

W 11:00

M 10:00 M 9:00

W 10:00

M 9:00

W 10:00 W 11:00

F 12:00

Football Marching Band

Color Guard

W 11:00

F 12:00

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CE MATH 1030

Midterm Review-KEY

Rubric 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)

Rubric 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)

9.

10.

a) 𝐸 = (0.1 ∗ 6) + (0.3 ∗ 1.50) + (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

a) 𝑦 = 29.99 + 0.69𝑚

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.

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CE MATH 1030

Midterm Review-KEY

15

10

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 2023.

1 pt. for correct answer and reasoning for part (e) 11.

12.

IBM

10 AT&T

5 GE

15

10

15

b) 10 + 15 = 25 people did not invest in either IBM or GE c) 15 people invested in just IBM d) 15 + 10 + 15 = 40 people invested in IBM or GE, but not AT&T

Rubric

1 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

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)

a) _{(2004−2009)}(16.3−12) = −4.2

5 = −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 = 4.2 million decrease

12 − 4.2 = 7.8 million sales c) 𝑦 = 16.3 − 0.84𝑥

d) Let 𝑦 = 0.

0 = 16.3 − 0.84𝑥𝑥 = 19.4

Sometime in 2023 sales of direct satellite receivers should be zero.

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CE MATH 1030

Midterm Review-KEY

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)

Rubric

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)

13.

14.

a) 𝐸 =(25∗125)+(30∗150)+(45∗175)+(15∗200)+(10∗225)_{25+30+45+15+10} =20740₁₂₅ = $166/day

b) 4550/26 = $175

c) $166_day ∗ 26 days = $4316

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) (45+15+10)/(25+30+45+15+10) = 70/125 = 0.56

a) 𝐴 = {𝑥: 𝑥 𝑖𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛_{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}

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CE MATH 1030

Midterm Review-KEY

Rubric 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) 15.

a) 𝑃(member of the crew) = 885

2201≈ 0.40

b) 𝑃(survive) = 771

2201≈ 0.35

c) 𝑃(survived given first class ticket) =203

325≈

0.62

d) 𝑃(third class ticket and survived) =

178

2201≈ 0.08