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MATH-138 Practice Problems (Unit 2)
1. A basketball player makes 80% of her free throws. Suppose she wakes up every
morning and starts shooting free throws. On an “average” morning, on which free throw
will she have her first miss? Perform 20 trials.
For this simulation we will generate random #’s from 1-5. Call any # from 1-4 a
“made shot” and a 5 a “missed shot”. For each trial, generate #’s until she “misses”
a shot. Note: the below trials were gotten using my calculator. If/when you repeat
this experiment, you will get different results (since we are dealing with random
#’s).
Trial 1: 5 (She missed on her first shot)
Trial 2: 1, 3, 3, 4, 1, 2, 5 (She missed on her sixth shot)
Trial 3: 2, 4, 5 (She missed on her third shot)
Trial 4: 2, 2, 1, 5 (She missed on her fourth shot)
Trial 5: 1, 1, 3, 5 (She missed on her fourth shot)
Trial 6: 5 (She missed on her first shot)
Trial 7: 2, 2, 1, 1, 4, 1, 3, 2, 5 (She missed on her ninth shot)
You would want to do at least 20 trials and then get the average # of times it took
her to miss. In the above example (with 7 trials), on average, she missed on her 28/7
= 4th shot.
2. You are going to take a quiz with 5 multiple choice questions. You estimate that you
have a 80% chance of getting any question right. What are you chances of getting them
all right? Perform 20 trials.
For this simulation we will generate random #’s from 1-5. Call any # from 1-4 a
“correctly answered question” and a 5 an “incorrectly answered question”. For each
trial, generate 5 #’s (corresponding to the 5 multiple choice questions). Note: the
below trials were gotten using my calculator. If/when you repeat this experiment,
you will get different results (since we are dealing with random #’s).
Trial 1: 5, 4, 2, 5, 4 (not all correct)
Trial 2: 2, 2, 5, 5, 1 (not all correct)
Trial 3: 1, 4, 5, 4, 2 (not all correct)
Trial 4: 1, 5, 1, 2, 1 (not all correct)
Trial 5: 2, 5, 2, 3, 4 (not all correct)
Trial 6: 1, 1, 1, 4, 4 (ALL correct)
Trial 7: 5, 2, 1, 3, 5 (not all correct)
Trial 8: 4, 3, 4, 5, 4 (not all correct)
Trial 9: 4, 3, 2, 2, 4 (ALL correct)
Trial 10: 5, 1, 3, 3, 2 (not all correct)
You would want to do at least 20 trials and then get the percentage of times that all
questions were answered correctly. In the above example (with 10 trials), all
questions were answered correctly 2/10 = 20% of the time.
3. You are going to take a quiz with 10 multiple choice questions, where each question
has 4 answer choices. You have not studied and you need to guess on each question.
What are your chances of passing the quiz (i.e. getting at least 6 out of 10 questions
correct on the quiz)? Perform 20 trials.
For this simulation we will generate random #’s from 1-4. Call any # from 1-3 an
“incorrectly answered question” and a 4 a “correctly answered question”. For each
trial, generate 10 #’s (corresponding to the 10 multiple choice questions). Note: the
below trials were gotten using my calculator. If/when you repeat this experiment,
you will get different results (since we are dealing with random #’s).
Trial 1: 4, 4, 2, 2, 1, 3, 3, 4, 4, 2 (4 correct questions . . . FAIL)
Trial 2: 4, 3, 2, 2, 4, 4, 2, 3, 2, 2 (3 correct questions . . . FAIL)
Trial 3: 1, 1, 2, 3, 1, 3, 1, 3, 2, 1 (0 correct questions . . . FAIL)
Trial 4: 4, 4, 2, 3, 1, 1, 3, 1, 3, 4 (3 correct questions . . . FAIL)
Trial 5: 4, 2, 1, 2, 3, 2, 4, 3, 1, 2 (2 correct questions . . . FAIL)
You would want to do at least 20 trials and then get the percentage of times that at
least 6 questions were answered correctly. In the above example (with 5 trials), the
student passed the quiz 0/5 = 0% of the time.
4. A company is downsizing and lays off three of its ten employees in one division. Sue,
John and Mary were let go. Each of them is over 55 years old and the three of them are
thinking of suing the company for age discrimination. Here is a breakdown of the ages of
the employees in their division:
Employee
Age
Bob
25
Charlie
33
Sam
35
Barbara
38
Ann Carol David Sue John Mary
48
54
58
57
56
64
Do a simulation to see if they have a case.
For this simulation we will generate random #’s from 1-10. Call any # from 1-6 a
“younger person getting let go” and a # from 7-10 an “older person getting let go”.
For each trial, generate 3 #’s (corresponding to 3 people being let go). Note: the
below trials were gotten using my calculator. If/when you repeat this experiment,
you will get different results (since we are dealing with random #’s).
Trial 1: 4, 3, 2 (Not all older)
Trial 2: 3, 8, 5 (Not all older)
Trial 3: 4, 8, 10 (Not all older)
Trial 4: 8, 2, 4 (Not all older)
Trial 5: 6, 6, 4 (Not all older)
Trial 6: 3, 3, 9 (Not all older)
Trial 7: 2, 6, 4 (Not all older)
Trial 8: 8, 10, 10 (ALL older)
Trial 9: 7, 7, 1 (Not all older)
Trial 10: 2, 8, 4 (Not all older)
You would want to do at least 20 trials and then get the percentage of times that all
older people are let go (assuming no discrimination). In the above example (with 10
trials), all older people were let go only 1/10 = 10% of the time. The spreadsheeet on
CourseCompass does this simulation with 10,000 trials and finds that only 6-7% of
the time all older people are let go. This is compelling evidence that the company
MAY have been employing age discrimination.
5. Suppose I come up with the following sampling scheme:
I will flip a fair coin and Jim, Shelby, Monique and Steve will be my sample if the coin
lands on heads; Monica, Jack, Karen and Sam will be my sample if the coin lands on
tails.
Suppose I flip the coin and it lands on heads (Jim, Shelby, Monique and Steve are my
sample).
Are they a random sample? Yes
Are they a simple random sample? No
6. Suppose I have a population of 4 people: Albert, Betsy, Chris and David. Suppose I
want to sample from this population and I derive the following sampling scheme: I put 3
pieces of paper into a hat. One piece of paper is labeled “1”, one piece of paper is labeled
“2”, and one piece of paper is labeled “3”. I will draw one piece of paper from the hat
(assume each piece of paper has an equal chance of being drawn). If the paper labeled
“1” is drawn, Albert and Betsy are my sample. If the paper labeled “2” is drawn, Chris is
my sample. If the paper labeled “3” is drawn, David is my sample. Will this sampling
scheme yield a random sample (Yes or No)? Yes
7. Suppose I have a population of 5 people: Albert, Betsy, Chris, David and Evelyn.
Suppose I want to sample 3 people from this population and I derive the following
sampling scheme: I will flip a fair coin twice and the results of the coin flips will
determine the sample:
Heads-Heads: Sample = Albert, Betsy, Chris
Heads-Tails: Sample = Albert, Betsy, David
Tails-Heads: Sample = Albert, Chris, Evelyn
Tails-Tails: Sample = Betsy, Chris, David
Will this sampling scheme yield a SIMPLE random sample (Yes or No)? No
8. Suppose we wanted to study how many credit hours HCC students are taking this
semester. How would we get a simple random sample (SRS) of HCC students? Stratified
sample? Cluster sample? Systematic sample? Convenience sample? Census?
See me if you have any questions on this question.
9. Which of the following (if any) are NOT valid probability values?
a. 0.40
b. -0.20
c. 1.00
d. 0.99999
10. Which of the following (if any) are NOT valid probability values?
a. 0.00
b. 0.67
c. 0.80
d. 1.14
Suppose I want to perform the random procedure of rolling a fair die (rolling it once).
Use this procedure to answer questions 11-17.
11. What is the sample space, S, for the above-mentioned procedure?
S = { 1, 2, 3, 4, 5, 6 }
Suppose the following events:
a. Rolling a 1
b. Rolling anything but 6
c. Rolling something less than 2
d. Rolling something between 2 and 5, inclusive (i.e. including 2 and 5)
12. What are the probabilities for the four events listed above?
a. 1/6
b. 5/6
c. 1/6
d. 4/6
13. What are the complements of the four events listed above?
a. Rolling a 2, 3, 4, 5 or 6
b. Rolling a 6
c. Rolling something 2 or greater
d. Rolling a 1 or 6
14. What are the probabilities of the four complements?
a. 5/6
b. 1/6
c. 5/6
d. 2/6
15. For the above die-rolling example, give an example of an impossible event.
Rolling a 7; Rolling a negative #, Rolling 2 #’s . . .
16. For the above die-rolling example, give an example of a certain event.
Rolling a # between 1 and 6; Rolling a # less than 37; Roll an even or odd #;
17. State whether each of the following pairs of events (from above) are mutually
exclusive or not:
a. Events a and b No
b. Events a and c No
c. Events b and d No
d. Events c and d Yes
The table below describes a standard deck of cards. Use the table to answer questions 1819. Note that I am counting aces as face cards.
Clubs (black)
Spades (black)
Hearts (red)
Diamonds (red)
Face Cards
4
4
4
4
Non-Face Cards
9
9
9
9
18. State whether each of the following pairs of events are mutually exclusive or not:
a. Black cards and red cards Yes
b. Black cards and diamonds Yes
c. Black cards and spades No
d. Diamonds and face cards No
e. Face cards and non-face cards Yes
f. Non-face cards and red cards No
19. Suppose I want to perform the random procedure of picking a card out of a standard
deck of cards. If ONE card is drawn, what are the following probabilities (please answer
using un-simplified fractions):
a. P(club) 13/52
b. P(not a heart) 39/52
c. P(face card) 16/52
d. P(red) 26/52
e. P(not a non-face card) 16/52
f. P(not black) 26/52
g. P(black or red) 52/52
20. One tie – dotted, striped, or solid – is selected at random, and then a shirt – white or
brown – is selected at random. What is the probability that a dotted tie AND white shirt
are selected?
a. 1/6
b. 1/2
c. 1/3
d. 3
e. None of these
21. What is the probability that you roll a die 4 times and get zero “6’s”? (5/6)^4=0.482
22. What is the probability that you roll a die 4 times and get at least one “6”?
1-((5/6)^4) = 0.518
23. What is the probability that someone who has 3 children has exactly one girl (assume
no twins or triplets)? 3/8
24. What is the probability that you flip a coin twice and get 2 tails? 1/4
25. What is the probability that you flip a coin 10 times and the seventh flip is heads? 1/2
The table below describes a standard deck of cards. Use the table to answer question 26.
Note that I am counting aces as face cards.
Clubs (black)
Spades (black)
Hearts (red)
Diamonds (red)
Face Cards
4
4
4
4
Non-Face Cards
9
9
9
9
26. Suppose I want to perform the random procedure of picking a card out of a standard
deck of cards. If ONE card is drawn, what are the following probabilities (please answer
using un-simplified fractions):
a. P(face card and black) 8/52
b. P(red or non-face card) 44/52
c. P(face card or not black) 34/52
d. P(black and red) 0
e. P(club or face card) 25/52
f. Given the card is black, what is P(club)? 13/26
g. Given the card is black, what is P(face card)? 8/26
h. Given the card is a non-face card, what is P(face card)? 0
Use the data below to answer question 27. This synthetic sample data (i.e. I made it up)
shows 1,000 people who either smoked or didn’t smoke, and who either died of lung
cancer or some other cause of death. Suppose I randomly sample one person from this
data.
Smoker
Non-Smoker
Lung Cancer Death
50
80
Non-Lung Cancer Death
150
720
27.
a. What is P(Smoker)? 200/1000
b. What is P(Lung Cancer Death)? 130/1000
c. What is P(Smoker given Lung Cancer Death)? 50/130
d. What is P(Non-Lung Cancer Death)? 870/1000
e. What is P(Non-Smoker)? 800/1000
f. What is P(Non-Lung Cancer Death given Non-Smoker)? 720/800
g. Is smoking and lung cancer death independent? No
28. Given P(A)=0.25, P(B)=0.60, and P(A and B)=0.10, find:
a. P(A or B) 0.75
b. P(B|A) 0.40
c. Are A and B independent (yes or no)? No
d. Are A and B mutually exclusive (yes or no)? No
Use the following information to answer questions 29: One tie – dotted, striped, or solid –
is selected at random, and then a shirt – white or brown – is selected at random.
29. What is the probability that a striped tie OR brown shirt is selected?
a. 1/2
b. 2/3
c. 1/6
d. 5/6
e. None of these
30. Suppose TWO fair dice are rolled. Find the following probabilities:
a. P(both die are 1) 1/36
b. P(the sum of the dice is 6) 5/36
c. P(at least one of the dice is 4) 11/36
d. P(only ONE of the dice is 4) 10/36
31. Suppose TWO fair dice are rolled. Let E be the event of getting a “triple” (i.e. one die
is three times the other die) and let F be the event of getting a “sum of 6” (i.e. the two
dice add up to 6). Which one of the following statements is true: P(E)>P(F), P(E)=P(F),
or P(E)<P(F)? P(E)<P(F)
32. Suppose a dresser drawer contains 20 individual socks where each sock is either
white or black (there is at least one of each color). Suppose you are blindfolded and you
start taking out socks from the drawer one by one. What is the MINIMUM number of
socks that you need to take out in order to GUARANTEE that you will have some
matching socks (i.e. 2 black socks OR 2 white socks). 3 Socks
33. For parts a-c below, state whether the pairs of events (events A and B) are dependent
or independent:
a. P(A)=0.60, P(B)=0.40, P(A and B)=0.24 Yes
b. P(A)=0.90, P(B)=0.30, P(A and B)=0.18 No
c. P(A)=0.50, P(B)=0.70, P(A and B)=0.25 No
34. Suppose a jar contains 40 red marbles, 40 blue marbles and 20 green marbles (100
marbles total). If TWO marbles are drawn WITHOUT REPLACEMENT from the jar
(that is, one marble is drawn and NOT put back into the jar, and then another marble is
drawn), what are the following probabilities?
a. P(both are green) (i.e. the first marble is green AND the second marble is green)
(20/100)*(19/99)
b. P(neither are green) (80/100)*(79/99)
c. P(first marble is red) (40/100)
d. P(first marble is red, second marble is blue) (40/100)*(40/99)
e. P(both marbles are neither red nor green) (40/100)*(39/99)
f. P(first marble is red, second marble is green) (40/100)*(20/99)
g. Given the first marble is red, what is P(second marble is red)? 39/99
h. Given the first marble is green, what is P(second marble is blue)? 40/99
35. If TWO cards are drawn WITH REPLACEMENT from a standard deck of cards (that
is, the first card is put back into the deck (and the deck is shuffled) before the second card
is drawn), what are the following probabilities?
a. P(both are black) (i.e. the first card is black AND the second card is black)
(26/52)*(26/52)
b. P(first card drawn is red, second card drawn is black) (26/52)*(26/52)
c. P(both cards are neither red nor face cards) (18/52)*(18/52)
d. P(first card drawn is a red face card, second card drawn is red) (8/52)*(26/52)
e. Given the first card drawn is a red card, what is P(second card is red)? 26/52
f. Given the first card drawn is a club face card, what is P(second card is a diamond face
card)? 4/52
36. Give the probabilities for questions #35A-F assuming the two cards are drawn
WITHOUT REPLACEMENT (that is, one card is drawn and NOT put back into the
deck, and then another card is drawn) and the deck is shuffled after replacement.
a. (26/52)*(25/51)
b. (26/52)*(26/51)
c. (18/52)*(17/51)
d. (8/52)*(25/51)
e. 25/51
f. 4/51
37. In Question #35, does the probability of the second draw depend on the first draw?
No
38. In Question #36, does the probability of the second draw depend on the first draw?
Yes
39. A large department store has 500 employees. There are 350 females and 200 of them
are under the age of 25. There are 75 males under 25. If one employee is randomly
selected, what are the following probabilities:
a. P(under 25 or female) 425/500
b. P(over 25 or female) 425/500
c. P(male or over 25) 300/500
40. There are 6 green hats, 4 blue hats and 3 red hats in a box. You randomly select one
hat. What are the following probabilities:
a. P(blue or red) 7/13
b. P(not green) 7/13
c. P(green or blue or red) 1
41. In a class of 50 students, 18 take chorus, 26 take band, and 2 take both. Answer the
following questions:
a. How MANY are only in chorus? 16
b. How many are only in band? 24
c. How many take neither? 8
d. How many take either band or chorus (but NOT both)? 40
42. Does the table below represent a valid probability distribution?
x
-3
-1.56
2
5.7
10,002
P(x)
0.20
0.10
0.05
0.56
0.09
Yes
43. Does the table below represent a valid probability distribution?
X
4
6
8
9
P(x)
-0.50
0.60
0.50
0.40
No
44. Does the table below represent a valid probability distribution?
X
0
1
No
P(x)
0.45
0.65
Suppose a random procedure that yields the following outcomes and probabilities. Use
this table to answer questions 45-46.
X
80
100
150
200
250
P(x)
0.24
0.22
0.31
0.18
0.05
45. Find the mean (expected value) and standard deviation of this distribution.
Mean=136.2; Std. Dev. = 49.9
46. What are the following probabilities:
a. P(X<=150) 0.77
b. P(X=200) 0.18
c. P(X<70) 0
d. P(X=100 or X=200) 0.40
e. P(X=100 and X=250) 0
f. P(X<200 or X>80) 1
47. Suppose I have a distribution where one third of the time the value equals -1, one
third of the time the value equals 0, and one third of the time the value equals 2. Is this a
valid probability distribution? Yes
48. Suppose I have a distribution where one half of the time the value equals 0.4, and two
thirds of the time the value equals 0.6. Is this a valid probability distribution? No
Use the following table to answer questions 49-50:
X
0
1
2
3
10
P(x)
0.0
0.3
0.3
0.3
0.1
49. Why is this probability distribution valid? The probabilities sum to one, and each
individual probability is between 0 and 1.
50. Find the expected value and standard deviation of this distribution.
Mean=2.8; Std. Dev. = 2.52
51. An insurance policy has the following pay offs. If you die, your survivor gets
$10,000. If you become disabled, you get $5000. Otherwise, you receive nothing. The
policy costs $50 a year. Based on past data, the probability a person dies is .01 and the
probability the person becomes disabled is .02. Find the expected value from the
company’s point of view.
-$150
52. A game costs $5 to play. You draw a card from a deck of cards. If you draw the ace
of hearts, you win $100. For any other ace, you get $10 and for any other heart you get
$5. If you draw anything else, you lose. Find the average winnings or losses for this
game.
-$1.35
53. A carnival game offers a $100 cash prize for anyone who can break a balloon by
throwing a dart at it. It costs $5 to play. You estimate that you have a 10% chance of
hitting the balloon on any throw. Find your expected winnings.
$5
54. A commuter must pass through 5 traffic lights on her way to work and will have to
stop at each one that is red. She estimates the probability model for the number of red
lights she hits as shown below.
X=# of red 0
1
2
3
4
5
P(x)
0.05 0.25 0.35 0.15 0.15 0.05
How many red lights should she expect to hit each day? 2.25
55. You roll a die. If it comes up 6, you win $100. If not, you get to roll again. If it
comes up 6 the second time, you win $50. If not, you lose. Create the probability model
and find the expected amount you’ll win. How much should you pay to play this?
The expected value is $23.61. Therefore, in theory, you should be willing to pay any
price less than this amount.
56. A man buys a racehorse for $20,000 and enters it in two races. He plans to sell the
horse afterwards hoping to make a profit. If the horse wins both races, it will sell for
$100,000. If it wins only one race, it will be worth $50,000. If it loses both races, it will
be worth $10,000. The man believes there is a 20% that the horse will win the first race
and a 30% chance that it will win the second race. Assuming the two races are
independent events, find the man’s expected profit.
$10,600
57. Suppose you visit Las Vegas and decide to play roulette. If you bet $5 that the
outcome is a number between 1-12 (including 1 and 12), you have a 26/38 probability of
losing your $5 bet, and you have a 12/38 probability of making a net gain of $10
(equaling the $15 prize minus your $5 bet). Only considering NET winnings/losses, what
is your expected value of betting on a number between 1-12 (round to the nearest cent)?
-$0.26
58. Suppose the following binomial probability situation: A certain statistics class has 15
students, and the probability that a given student will pass the class is 0.8. Find the
following probabilities:
a. P(everybody passes) 0.035
b. P(at least 10 students pass) 0.939
c. P(4 students fail) 0.188
d. P(11 or 12 students pass) 0.438
e. P(at most 2 students fail) 0.398
59. Suppose the following binomial probability situation: You draw a card out of a
shuffled deck of cards 10 times (replacing the card after each draw and re-shuffling) and
count the number of red cards you draw (note there are 26 red cards and 52 cards total).
Find the following probabilities (3 decimal places):
a. P(6 reds) 0.205
b. P(3 blacks) 0.117
c. P(at most 5 reds) 0.623
d. P(more than 7 blacks) 0.055
60. A moving target at a police academy target range can be hit 80% of the time by a
particular individual. Suppose the person takes three shots at the target. What is the
probability that:
a. There are exactly two hits? 0.384
b. There are hits on all three? 0.512
c. There is only one hit? 0.096
d. There are misses on all three? 0.008
e. There is at least one hit? 0.992
61. Suppose that 40% of all college students smoke cigarettes. If 15 are selected at
random:
a. What is the probability that 10 smoke? 0.024
b. What is the probability that at most 12 of the students smoke? 0.9997
c. What is the probability that between 5 and 11 students smoke? 0.781
62. A quality control inspector has drawn a sample of 13 light bulbs from a recent
production lot. If the number of defective bulbs is 2 or less, the lot passes inspection.
Suppose 10% of the bulbs in the lot are defective. What is the probability that the lot will
pass inspection? 0.866
63. Suppose the following binomial probability situation: Suppose Dr. Coldren was a
single (i.e. not married), heterosexual male. Further suppose that there was a week in the
distant past (Sunday-Saturday) where he asked a different supermodel for a date (for that
evening) each day of the week. Suppose the probability that any given supermodel said
“yes” was 0.20. Assume a supermodel agreeing to a date was a “success”, and not
agreeing to a date was a “failure” (meaning I stayed home alone for the evening). Find
the following probabilities:
a. P(Dr. Coldren stayed home alone all week) 0.210
b. P(Dr. Coldren had dates with supermodels only on days that began with “S”) 0.275
c. P(Dr. Coldren stayed home alone at least one evening) 0.9999872
d. P(Dr. Coldren had a date with a supermodel every evening of the week) 0.0000128
e. P(Dr. Coldren was home alone an odd number of evenings) 0.514
64. Suppose 2000 subjects are asked to select whether Pepsi or Coke tastes better. If it is
assumed that there is no difference in product preference, what is the probability of
observing 900 or less subjects who thought Coke was superior? 0.000004
65. Suppose a virus is believed to infect two percent of the population. If a sample of
3000 subjects are tested, what is the probability that:
a. More than 30 of the subjects will be infected? 0.999988
b. Between 40 and 80 subjects will be infected? 0.9924
c. At least 70 of the subjects will be infected? 0.1095