Download Problem Set Section 3.1 Probability Basics Indentifying Probabilities

Problem Set Section 3.1
Probability Basics
1. Indentifying Probabilities.
a. What is the probability of an event that is certain to occur?
b. What is the probability of an impossible event?
c. On a true/false test, what is the probability of answering a question correctly if
you guess?
d. On a multiple choice test with four possible answers for each question, what is
the probability of answering a question correctly if you guess?
2. Experiment: Pick one card for a standard deck of playing cards
a. Event A: observe an Ace ; find P(A)
b. Event B: observe a heart ; find P(B)
c. Event C: observe a face card ; find P(C)
d. Event D: observe a black card; find P(D)
3. Experiment: Roll a dice once
a. Event A: observe a three ; find P(A)
b. Event B: observe an even number ; find P(B)
c. Event C: observe a prime number ; find P(C)
4. Fruit: If there are 6 apples, 3 oranges, and 1 plum in a basket, what is the probability of
choosing an apple without looking in the basket?
a. Define the experiment
b. Define the event
c. How many outcomes are favorable to the event?
d. How many total outcomes are there?
e. Find the probability of the event?
5. Gender of Children: Suppose a couple is going to have 2 children. Assume the
probability of a boy or girl is the same (this isn’t necessarily true).
a. Define the experiment
b. List the sample space
c. What is the probability of having exactly one girl and one boy?
d. What is the probability of having all boys?
6. Flipping Coins: Suppose a coin is flipped 3 times.
a. Define the experiment.
b. List the sample space
c. What is the probability of getting exactly one tail and two heads?
d. What is the probability of getting three tails and no heads?
1
7. Dice Probabilities: Roll two dice and observe the sum of the dice.
a. Define the experiment.
b. List the sample space.
c. What is the probability of rolling a seven?
d. What is the probability of rolling a three?
e. What is the probability of rolling a two?
f. Are any of these probabilities unlikely?
8. Dice Probabilities: Roll two dice and observe 4’s.
a. How many elements are in the sample space?
b. Find P(no 4’s), P(one 4), P(two 4’s)
c. What is the sum of the probabilities in b? Can you explain this?
9. Probability of a Birthday:
a. If an individual is selected at random, what is the probability that their birthday
is July 1st. Ignore leap years.
b. If an individual is selected at random, what is the probability that their birthday
is in July. Ignore leap years.
c. If an individual is selected at random, what is the probability that their birthday
is not in July. Ignore leap years.
10. Compliments: Suppose, for example, the probability that Bill will graduate from college
is 0.80. What is the probability that Bill will not graduate from college?
11. Stolen Cars: According to a study, the three cars most frequently stolen are listed in the
table.
Car Make
Camaro
Honda Accord
Toyota Camry
Number
Stolen
287
256
195
If one stolen car from the table is randomly selected, find the probability that the car is a
Camaro? An Accord? A Camry? What is the sum of all the probabilities? What can you
conclude?
12. Stolen Cars: Using the data from number 11, what is the probability that the car stolen
is not an Accord?
2
13. Cheating: In a survey of college students, 876 said that they have cheated on an exam
and 1765 said that they have not.
a. If one college student is select at random, find the probability that the student
has cheated.
b. Find the probability of the compliment of the event defined in part a.
c. Are either of these probabilities unlikely? Why or why not?
14. Adverse Drug Reaction: In clinical tests of a blood pressure reduction drug 85 patients
reported dizziness and 330 did not. Based on this sample, what is the probability that a
user of this drug will experience dizziness? Is the probability high enough for the drug
company to be concerned?
15. Roulette: In the game of roulette there are 38 numbers, 18 numbers are colored red,
and 18 numbers are colored black. Additionally there are 2 green colored numbers 0
and 00 for a total of 38 numbers.
a. What is your probability of a 2 or 10?
b. What is the probability of an even?
c. What is the probability of a 1 through 12?
d. What is the probability of a red?
3
Problem Set Section 3.2
Addition Rule
1. Decide whether events are disjoint:
a. A die is rolled
i. Observe an even number
ii. Observe an odd number
b. A card is drawn from a deck of cards
i. A seven is observed
ii. A heart is observed
c. A student is selected
i. The student is taking math
ii. The student is a business major
d. A politician is selected
i. The politician serves in Congress
ii. The politician is governor of Hawaii
2. A bag contains 4 white marbles, 6 black marbles, and 1 green marble. What is the
probability of drawing:
a. A white marble?
b. A black marble?
c. A green marble?
d. A marble that is not white?
e. A marble that is not black?
f. A marble that is not green?
g. A white marble or a black marble?
h. A white marble or a green marble?
i. A black marble or a green marble?
3. A card is drawn randomly from a deck of ordinary playing cards. You win $10 if the card
is a spade or an ace. What is the probability that you will win the game?
4. A student goes to the library. The probability that she checks out (a) a work of fiction is
0.40, (b) a work of non-fiction is 0.30, and (c) a work that is both fiction and non-fiction
is 0.20. What is the probability that the student checks out a work of fiction or a work of
non-fiction or both?
4
5. Smoking and Heart Disease: A study was done to determine to compare the rate of
heart disease in people who smoke vs. those that don’t smoke. The results are listed
below.
Heart Disease
Yes
No
Total
Smoke
37
13
50
Don't Smoke
6
144
150
Total
43
157
200
a. P(Smoke)
b. P(Heart Disease)
c. P(Smoke or Heart Disease)
d. P (Heart Disease or Don’t Smoke)
6. Vitamin C and Colds: Does vitamin C reduce the incidence of colds? To test this
suppose we asked 100 people two questions: whether they take vitamin C tablets, and
whether they had a cold last year.
Vitamin C
Had a cold?
Yes
No
Total
No
35
5
40
Yes
35
25
60
70
30
100
a. P(Cold)
b. P(No Vitamin C)
c. P(Cold or No Vitamin C)
d. P (Cold or Vitamin C)
7. Determine Whether Events are Disjoint:
a. P(A) = 3/7, P(B) = 2/7, P(A or B)=5/7, what do you know about events A and B?
Explain.
b. P(A) = 8/25, P(B) = 7/25, P(A or B)=12/25, what do you know about events A and
B? Explain. What is the P(A and B)?
c. If A and B are mutually exclusive and P(A) = 0.3, P(B) = 0.2 what is P(A and B)?
5
Problem Set Section 3.3
Multiplication Rule
1. Decide whether events are independent or dependent:
a. Toss a coin and observe a head
Roll a die and observe a 5
b. Playing golf
Going to work
c. Your car doesn’t start
Finding out you’re out of gas
d. Your car doesn’t start
Your wearing flip flops
2. What is the probability that a fair coin tossed 4 times comes up heads each time? What
is the probability that a fair coin tossed 10 times comes up tails each time? Are either of
these probabilities unlikely? Explain.
3. A fair coin is tossed 10 times, and comes up heads each time. What is the probability
that on the 11th toss, it comes up heads again?
4. Find the probability of flipping a coin and observing a head and rolling a die and
observing 5.
5. Geometric Probability: A fair, six-sided die is tossed. What is the probability the first 5
occurs on the fourth roll?
6. Drawing Marbles: An urn contains 6 red marbles and 4 black marbles. Two marbles are
drawn.
a. What is the probability that both of the marbles are black if the first marble is
replaced?
b. What is the probability that both of the marbles are black if the first marble is
not replaced?
7. What is the probability that two randomly selected people were born on the same day?
Ignore leap years.
8. Guessing on a True / False Test: A 10 question true / false test is given.
a. What is the probability of guessing all 10 correct?
b. What is the probability of guess the first 7 questions correct and the last 3
incorrect?
c. If 70% is a passing grade, is this the only way of guessing questions and still
passing?
6
9. Guessing on a Multiple Choice Test: A 10 question multiple choice test is given with 5
possible answers for each questions.
a. What is the probability of guessing all 10 correct?
b. What is the probability of guess the first 7 questions correct and the last 3
incorrect?
c. If 70% is a passing grade, is this the only way of guessing questions and still
passing? Explain.
10. Excuses: Four students missed a test because they said their car had a flat tire. To see if
they were telling the truth the teacher put a question on the make up test asking them
which tire went flat. If they really didn’t have a flat tire, what is the probability that they
all guessed the same?
11. Vitamin C and Colds: Does vitamin C reduce the incidence of colds? To test this
suppose we asked 100 people two questions: whether they take vitamin C tablets, and
whether they had a cold last year.
Vitamin C
Had a cold?
Yes
No
Total
No
35
5
40
Yes
35
25
60
70
30
100
a. If two subjects are randomly selected, find the probability that they both
took vitamin C.
b. If two subjects are randomly selected, find the probability that they both got
a cold.
c. If three subjects are randomly selected, find the probability that they all did
not take vitamin C.
d. If one is selected find the probability of getting someone who took Vitamin C
OR got a cold.
e. If one is selected find the probability of getting someone who did not take
Vitamin C OR got a cold.
f. Find the probability of getting someone who took Vitamin C AND got a cold.
i.
ii.
Read directly from the table
Apply the multiplication rule
g. Find the probability of getting someone who did not take Vitamin C AND got
a cold.
i.
ii.
Read directly from the table
Apply the multiplication rule
7
12. Beer Drinkers: Consider a sample of 200 beer-drinkers. For each drinker we have
information on sex ("Male" and "Female") and preferred category of beer ("Light",
"Regular", "Dark"). A contingency table for these data might look like the following.
Light
Regular
Dark
Total
Male
20
40
50
110
Female
50
20
20
90
Total:
70
60
70
200
a. If two subjects are randomly selected, find the probability that they both
drink light beer.
b. If two subjects are randomly selected, find the probability that they both are
female.
c. If one is selected find the probability of getting a female OR someone who
drinks regular beer.
d. If one is selected find the probability of getting a male OR someone who
drinks dark beer.
e. Find the probability of getting a female AND someone who drinks regular
beer.
i.
Read directly from the table
ii.
Apply the multiplication rule
f. Find the probability of getting a male AND someone who drinks dark beer.
i.
Read directly from the table
ii.
Apply the multiplication rule
13. Sex of Babies: Do certain families have a tendency to have babies of the same sex?
Let's assume that the sexes of babies are independent, and that the probability that a
baby is born female is 0.5014, which is approximately the current percentage. Assume
that this probability is the same regardless of family size (which is an assumption that is
not necessarily true).
a. What is the chance that a family with 6 children has them born in alternating sex
order, with the oldest being a male?
b. What is the chance that a family has 6 girls in a row?
c. Given that the first three children are boys, what is the chance that the next
child will be a girl?
8
14. Blood Types: Blood comes in four types: O, A, B, and AB. The percentages of people in
the United States with each blood type are shown below.
Blood Type
O
A
B
AB
Percentage
46
40
10
4
a. What is the probability that two people getting married both have blood type
O? What assumption are you making?
b. What is the probability that two people getting married both have the same
blood type? What assumptions are you making?
15. Determine Whether Events are Independent:
a. P(A) = 3/7, P(B) = 2/7, P(A and B)=6/49, what do you know about events A and B?
b. P(A) = 8/25, P(B) = 7/25, P(A and B)=15/625, what do you know about events A
and B?
c. If A and B are independent and P(A) = 0.3, P(B) = 0.2 what is P(A and B)?
9
Problem Set Section 3.4
Compliments and Conditional Probability
1. Write the complement of the event.
a. A student guesses on a 10 question multiple choice test, he gets at least one
question correct.
b. A batch of 1000 CD’s are shipped, all are free of defects.
c. Roll a die 5 times, no four is rolled.
d. A couple decided to have 4 children, they have at least one girl.
2. At Least One Correct Answer: A 10 question multiple choice test is given with 5 possible
answers for each questions.
a. What is the probability of guessing all 10 correct?
b. What is the probability of guessing none correct?
c. What is the probability of guessing at least one correct?
3. At Least One Head: For fun on Saturday night, you and a friend are going to flip a fair
coin 10 times (geek!). Because the coin is fair, assume P(Head) = P(Tail) = 0.5. Neither
of you know how to flip the coin to obtain some desired outcome.
You flip
Your friend flips
HTHHTHTTTH.
HHHHHHHTTT.
a. Which sequence is more likely to occur?
b. What is the probability of getting no heads?
c. What is the probability that you will get at least one heads in ten flips?
4. At Least One Number: Roll a die 2 times
a. What is the probability of rolling a four on each roll?
b. What is the probability of not rolling a four on either roll?
c. What is the probability of rolling at least one four on the 2 rolls?
5. Famous Birthday Problem:
Three people are randomly selected
a. What is the probability that no two share the same birthday?
b. What is the probability that at least two share the same birthday?
6. Let P(A)=1/2, P(B)=1/3, and the
a. Find
b. Find
c. Find
10
7. Let P(A)=3/8, P(B)=5/8, and the
a. Find
b. Find
c. Find
8. Probability of a Boy: What is the probability of a couple having a baby girl when their
third child is born, given that the first two were boys? Is the result the same as the
probability of having three girls?
9. Failing Students: In a certain college, 25% of the student failed mathematics, 15% of
the students failed chemistry, and 10% failed both mathematics and chemistry. A
student is selected at random.
a. What is the probability that they failed mathematics given they failed chemistry?
b. What is the probability that they failed chemistry given they failed mathematics?
c. What is the probability then they failed mathematics or chemistry?
10. Vitamin C and Colds: Does vitamin C reduce the incidence of colds? To test this
suppose we asked 100 people two questions: whether they take vitamin C tablets, and
whether they had a cold last year.
Vitamin C
Had a cold?
Yes
No
Total
No
35
5
40
Yes
35
25
60
70
30
100
a. Find the probability of getting a cold, given Vitamin C.
b. Find the probability of not getting a cold, given Vitamin C.
c. Find the probability of getting a cold, given no Vitamin C.
d. Find the probability of not getting a cold, given no Vitamin C.
11
11. Grades of Students: The following is a table of grades received in statistics classes
between men and women.
Men
Women
TOTAL
A
14
24
38
B
24
36
60
C
24
34
58
D
12
20
32
F
6
6
12
TOTAL
80
120
200
P(B)
P(M and C)
P(M|B)
P(A or B)
P(C and M)
P(B|W)
P(A or M)
P(A and B)
P(A|W)
P(A)
P(B and M)
P(B|M)
12
Problem Set Section 3.5
Counting Techniques
1. Calculate the following
a. 6!
b) 15!
c) 25P4
d) 100P3
e) 25C4
f) 100C3
2. A business man has 4 dress shirts and 7 ties. How many different shirt/tie outfits can he
create?
3. How many sample points are in the sample space when a coin is flipped 4 times?
4. How many sample points are in the sample space when a die is rolled 3 times?
5. How many different ways can you arrange the letters X, Y, and Z?
6. Five different books are on a shelf. In how many different ways could you arrange
them?
7. How many different 3-digit numerals can be made from the digits of 45678 if a digit can
appear just once in a numeral?
8. In a conference of 9 schools, how many intra conference football games are played
during the season if the teams all play each other exactly once?
9. How many different ways can you select 2 letters from the set of letters: X, Y, and Z?
a. If order matters?
b. If order doesn’t matter?
10. Given the letters A, B, C, D, E, F.
a. How many ways can they be arranged if duplicates are not allowed?
b. How many ways can you choose 3 of the letters if duplicates are allowed?
c. How many ways can you choose 3 of the letters if duplicates are not allowed and
order matters?
d. How many ways can you choose 3 of the letters if duplicates are not allowed and
order doesn’t matter?
11. Horse Racing: In horse racing, a trifecta is a type of bet. To win a trifecta bet, you need
to specify the horses that finish in the top three spots in the exact order in which they
finish. If eight horses enter the race, how many different ways can they finish in the top
three spots?
12. Selected Students: A teacher has 15 students and 5 are to be chosen to give
demonstrations. How many different ways can the teacher choose the demonstrators
given the following conditions?
a. The order of the demonstrators is important?
b. The order of the demonstrators is not important?
13
13. Lottery: A state lottery is played by selected 6 numbers out of 49. Find the probability
that all six of numbers you selected come up if
a. Order doesn’t matter
b. Order matters
14. Access Codes: The access code to a houses security system consists of five digits.
a. How many different codes are available if each digit can be repeated?
b. How many different codes are available if the first digit cannot be zero and the
arrangement cannot be all the same digit?
15. Drawing Marbles: 3 marbles are drawn at random from a bag containing 3 red and 5
white marbles.
a. How many different draws are there?
b. How many different draws would contain only red marbles?
c. How many different draws would contain 1 red and 2 white marbles?
d. How many different draws would contain exactly 2 red marbles?
16. Drawing Students: Eight student names will be drawn at random from a hat containing
14 freshmen names, 15 sophomore names, 8 junior names, and 10 senior names.
a. How many different draws of 8 names are there overall?
b. How many different draws of 8 names would contain only juniors?
c. How many different draws of 8 names would contain exactly 4 juniors and 4
seniors?
17. Group Project: There are 12 men and 14 women in Professor Clarkson’s math class.
Find the number of ways Professor Clarkson can select a team of 3 students from the
class to work on a group project. The team consists of 1 girl and 2 boys.
18. Gender of Children: Ten newborn babies are selected at random.
a. How many different possibilities are there?
b. How many ways can 4 boys and 6 girls be arranged?
c. What is the probability of selecting 4 boys?
d. Is it common to get 4 boys when selecting 10 babies?
19. Famous Birthday Problem:
Twenty-five people are randomly selected
a. What is the probability that no two share the same birthday?
b. What is the probability that at least two share the same birthday?
14
Selected Answers:
Section 3.1
1a) 1
1b) 0
1c) 1/2
1d) ¼
2a) 4/52
2b) 13/52
2c) 12/52
2d) 1/2
3a) 1/6
3b) 3/6
3c) 3/6
4a) choosing an apple
4b) observing an apple 4c) 6
4d) 10
5a) couple having 2 children
5b) {GB, BG, BB, GG}
5c) ½
5d) 1/4
6c) 3/8
6d) 1/8
7a) roll 2 die
7c) 6/36
7d) 2/36
7e) 1/36
8a) 36
8b) 25/36, 10/36, 1/36
11)P(Co)=287/738
P(A)=256/738
P(Cy)=195/738
P(All)=1
12) 482/738
13a) 876/2641
13b) 1765/2641
13c) Neither
14)85/415
4e) 6/10
Section 3.2
1a) disjoint
2d) 7/11
4) 0.5
5a) ¼
7a) disjoint
1b) not disjoint
2h) 5/11
5b) 43/200
5c) 56/200 = 7/25
7b) not disjoint, 3/25
5d) 187/200
7c) 0
Section 3.3
1a) independent1
2) 0.0625 & 0+
3) 0.5
4) 0.0833
5) 0.0965
6a) 0.16
8a) 1/1024
9a) 1/9765625
10) 1/64
11a) 0.488
12b) 0.201
13a) 0.0156
1c) dependent
6b) 0.133
8b) 1/1024
9b) 64/9765625
8c) no
9c) no
11c) 0.0251
12d) 0.65
13b) 0.0159
11d) 0.95
12e) 0.1
13c) 0.5014
15
11g) 0.25
14a) 0.212
14b) 0.3832
Section 3.4
1b) A batch of 1000 CDs are shipped and at least one is defective.
2a) 0+
2b) 0.107
2c) 0.893
3b) 0+
3c) 0.999
4a) 0.0278
4b) 0.694
4c) 0.306
5a) 0.992
5b) 0.00820
6a) 3/4
6b) 1/2
6c) 7/12
7a) 1/4
7c) 2/3
9a) 0.667
9b) 0.4
9c) 0.3
10a) 1/2
10b) 1/2
10c) 5/6
10d) 1/6
#11
P(A)=38/200
P(A or B)=P(A)+P(B)=98/200
P(B and M)=24/200
P(C and M)=24/200
P(A and B)=0
P(B|M)=24/80
P(M|B)=24/60
P(A|W)=24/120
Section 3.5
5) 3!
6) 5!
10a) 720
10b) 216
12b) 3003
15a) 8C3
16a) 314,457,495
17) 12C2 x 14C1
18a) 1024
7) 5P3
10c) 120
15b) 3C3
16b) 1
18b) 210
8) 9C2
10d) 20
15c) 3C1 x 5C2
16c) 8C4 x 10C4
19a) 365P25/36525
16