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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)
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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
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