Module 7: Submitted Homework Assignment
**Module 7 Submitted Homework Assignment is worth 20 points (5% of your grade)
Write answers for the following and submit them via email according to the schedule in the
course syllabus. Be sure your answers are contained in the body of your message. Do NOT send
them as attachments.
Send your answers to Module 7 to the instructor: mccartc1@ohio.edu
The questions are structured so that a single letter, word, or number will suffice. Computational
questions are arranged so that partial credit can be given for each step answered correctly.
Always use the following model to submit your answers to the questions.
EXAMPLE:
Your name:
Module Number:
Answers
Q1 C
Q2 B
Q3 A, etc.
If the question requires computation, do the calculations and then give the correct
values as follows:
(Always hold all decimal values through your computations, and round final
answers to at least two decimal places)!
Q4 7
Q5 4
Q6 22, etc.
If the question is a fill in the blank, just put in the appropriate word(s) as follows:
Q7 statistics
Q8 dependent variable, etc.
Module 7 Questions—Submit answers via email to mccartc1@ohio.edu according to above
instructions: (There are 20 questions to be answered for Module 7)
The following 6 questions (Q1 to Q6) are based on the following summarized data below:
Given the upcoming NBA draft, there are 100 players available:
Point Guard (PG)
Shooting Guard (SG)
Center (C)
Small Forward (SF)
Power Forward (PF)
College Experience (CE)
15
17
10
17
16
No College Experience (NCE)
3
8
8
2
4
Find the following probabilities:
Q1: p(PF)
Q2:
p(C and NCE)
Q3:
p(CE)
Q4:
p(SF/CE)
Q5:
p(not SG)
Q6:
p(CE/PF)
The following 3 questions (Q7 to Q9) are based on the following information:
You have a kennel that contains all Labrador Retrievers. The kennel has 6 black labs, 3 yellow
labs, and 1 chocolate lab.
Find the following probabilities:
Q7: What’s the probability that you randomly select a black lab, don’t put the lab back, then
draw another black lab, don’t put the lab back, and then draw another black lab?
Q8: What’s the probability that you draw a yellow or chocolate lab on the first draw, replace
the dog, and then draw a black lab?
Q9: What’s the probability that you draw a yellow lab on the first draw, don’t put the lab
back, and then draw either a yellow or chocolate lab on the second draw?
The following 3 questions (Q10 to Q12) are based on the following information:
John has the phone numbers of many young ladies. Some of these young ladies are
blonde and others are brunettes. In addition, John knows that some are single while others have
boyfriends. The exact breakdown is as follows:
Blonde
Brunette
Single
15
18
Taken
10
12
Find the following probabilities:
Q10: What is the probability of selecting someone who is taken, given that they are blonde?
Q11: What is the probability of selecting a single brunette?
Q12: What is the probability of selecting someone who is brunette, given that they are taken?
Q13: A coin is tossed four times. Which of the following sequences of heads (H) and tails(T) is
more likely?
A. HTHT
B. HHHT
C. THHH
D. TTTT
E. all are equally likely
F. none of the above
Q14: If an event can occur twice out of every 10 times, it has the probability value of
A. 0.10
B. 0.20
C. 0.50
D. none of the above three answers are correct
Q15: Probabilities are expressed with what values?
A.
negative infinity to positive infinity
B.
0.00 to positive infinity
C.
0.00 to +1.00
D.
–1.00 to +1.00
The following 5 questions (Q16 to Q20) are either “True” or “False”
Q16: The ‘Subjective View’ uses an analysis of possible outcomes to define probability
Q17: ‘Gambler’s Fallacy’ is the incorrect belief that the probability of a particular event
changes after a series of the event has taken place.
Q18: ‘Mutually Exclusive Events’ is when the occurrence of one event has no effect on the
probability of occurrence of the other.
Q19: The probability of randomly guessing at the first three multiple choice questions on an
exam, each of which has four possible answers, is 0.015625.
Q20: Flipping a coin repeatedly is a series of “Independent Events’
MODULE 7
Lesson 7: Probability
TEXTBOOK REFERENCES:
1. Howell—Chapter 7 Basic Concepts of Probability
2. Utts—Chapter 16 Understanding Probability and Long-Term Expectations
3. Utts—Chapter 17 Psychological Influences on Personal Probability
4. Utts—Chapter 18 When Intuition Differs from Relative Frequency
COURSE NOTES:
Probability
-You may wonder why probability is being covered in this course?
-probability is an important foundation for inferential statistics.
-very useful in hypothesis testing
-the prediction about a population based on sample data results
-therefore, probability helps to assess the generalizability of data analysis
-We use probability in our daily lives
a)
weather—chance of rain is 0.20 vs 0.90
-when to carry an umbrella?
b)
sports—basketball: late in a game, deliberate foul to keep opponent from scoring.
-Further, who do you foul? Shaquille O’Neal (0.60) vs Kobe Bryant (0.80). That is, foul
the player with the lower probability of making the free throw.
-However, sometimes we use probability incorrectly
a)
Gambler’s Fallacy:
-the incorrect belief that the probability of a particular event changes after a series of the
event has taken place.
Example #1 -toss a coin, call heads or tails,
p (heads) = 0.50
-toss a coin 15 times. What is the probability of obtaining heads on the
16th coin toss,
p (heads) is still = 0.50
-flip the coin 100 times,
p (heads on the 101st flip)=0.50
-flipping the coin repeatedly is a series of independent events. The result of a coin toss is
NOT related to the outcome of the previous toss.
General Definition of Probability:
-The likelihood of something occurring
-When several different outcomes are possible, the probability for a particular outcome, A, is defined as:
p(A) = # of outcomes classified as A
total # of possible outcomes
PSY1110: Module 7 Course Notes
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Examples:
1)
2)
-deck of cards (standard deck of 52 cards)
-possible outcomes (52)
p(a spade)
= 13/52 = 1/4 = 0.25
p(2)
= 4/52 = 0.077
p(king of hearts)
= 1/52 = 0.019
-term used in all sorts of ways, so let’s get them straight before we go on.
Probability:
I)
-toss one die
-possible outcomes (1,2,3,4,5,6)
p(1)
= 1/6 = .167
p(even number)
= 3/6 = 0.50
p(getting number less than or equal to 5)
= 5/6 = 0.83
Views of Probability:
1)
Analytic:
-an analysis of possible outcomes to define probability
-Bag contains 1 black marble and 9 green marbles. What is the probability of my
drawing a black marble?
= 1/10 = 0.10
p(A) =
A
A+B
-if event could be A
or B and all events equally likely
p(boy baby) = 1/2 = 0.50
=1 choice / 2 possibilities
2)
II)
Relative Frequency View:
-defines probability in terms of past performances/outcomes.
--waiting for a bus and after many, many days of doing this you notice that 75/100 times
the bus is late.
p(bus late) = 0.75
3)
Subjective View:
-not based on actual numbers or calculations: probability defined in terms of personal
belief in a likely outcome
*may not be accurate, but it is important, because it influences our behavior
-weather (play golf or not)
-likelihood of being punished if committing a crime
Some Basic Terminology:
a)
Event
-the outcome of a trial (could be the data)—any kind of outcome
-the basic bit of data—the ‘thing’ whose probability we are calculating
b)
Independent Events
-events are independent when the occurrence of one event has no effect on the
probability of occurrence of the other
-2 things that have no influence on each other, nothing to do with each other
*dice rolls
PSY1110: Module 7 Course Notes
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III)
c)
Mutually Exclusive Events
-the occurrence of one event precludes (makes impossible) the occurrence of the
other
*eg: freshman, sophomore, junior, senior
*eg: Gender; days of the week
--if it is one, it cannot be the other
d)
Exhaustive
-the set of events that represents all possible outcomes
e)
Sample with Replacement
-sampling in which the item drawn of trial N is replaced before the drawing on
trial N + 1 (the next trial)
f)
Sample without Replacement
-the item drawn is not replaced before the subsequent trial
Basic Laws of Probability:
1)
-Nonnegative
-Between 0 and 1, including 0 and 1
0 < p(A) < 1
-for any event (A), the probability of A occurring is between 0 and 1
-if an event never occurs p(event) = 0
ie: p(tail) on a two-headed coin
-if event always occurs p(event) = 1
ie: p(head) on a two-headed coin
2)
Addition Law (Addition Rule)
-used when probability of two or more simple events is desired
-for mutually exclusive random events, the probability of either one or the other
occurring equals the sum of the probabilities of each simple (individual) event.
(Mutually exclusive means there are no common outcomes)
p(A or B) = p(A) + p(B)
Examples:
a)
b)
c)
d)
PSY1110: Module 7 Course Notes
p(1 or 6) in die toss
=1/6 + 1/6
=2/6
=0.33
p(H or T) in coin flip
=1/2 + 1/2
=1
p(Ace or King) in cards
= 4/52 + 4/52
= 8/52
= 0.15
-in rolling a fair die once, what is the probability or rolling a one or
an even number
p(1 or even #)
Page 3
= p(1) or p(even #)
=1/6 + 3/6
=4/6
=0.67
e)
Randomly sample one person from a population of 100 people: 20 children, 30
teens, and 50 adults.
-What is the probability that your selectee is a teen or adult?
p(teen or adult) = p(teen) + p(adult)
= 30/100 + 50/100
= 80/100
=0.80
….is a child or teen?
= 20/100 + 30/100
=0.50
**if not mutually exclusive events**
p(A or B) = p(A) + p(B) – p (A and B)
-the last term removes the possibility of them occurring together
Example:
p(King or heart)
= (4/52 + 13/52) – (4/52)(13/52)
= (0.0769 + 0.25) – (0.0769*0.25)
= 0.3269 – 0.0192
= 0.3077
-count them:
= 13 hearts + 3 (other) kings
= 16 cards/52 total
= 0.3077
(get same answer)
3)
Multiplication Rule
-if 2 events are independent, the probability of both of them occurring together is
the product of their separate probabilities
= p(A) * p(B)
**remember: independent events means the occurrence of one has no effect on
the other
**where as additive rule gives p of any one of several events, multiplication rule
is concerned with joint or successive occurrence of several events.
Example:
-flip two coins; what is probability they both come up heads?
p(both coins coming up heads)
=p(head, head)
=p(head) * p(head)
=1/2 * 1/2
=0.25
PSY1110: Module 7 Course Notes
Page 4
-roll a pair of fair die once; What is probability of 2 on first die, and a 4 on second die)?
p(2 on die 1, 4 on die 2)
=1/6 * 1/6
=0.0278
-suppose you are randomly sampling 2 people from a population of 110: 50 men, 60
women (sampling is one at a time with replacement)
p(woman on 1st, woman on 2nd)
=60/110 * 60/110
=0.545 * 0.545
=0.297
**why is “with replacement” so important? Because if you DON’T replace, events are
not independent
p(woman, woman)
= 60/110 * 59/109
= 0.545 * 0.541
= 0.295
IV
Types of Probability (aside from a single event)
a)
Joint: -the probability of the co-occurrence of 2 or more events
p(A, B)
*if independent, can use the multiplication rule to compute.
b)
Conditional Probability:
-the probability that one event will occur GIVEN that some other event has occurred
*denoted p(A / B):
-the probability of A, given B
-the probability of A, if B is true
(these are non-independent events)
Example:
Take a survey of political party affiliation and Bush’s tax-reform plan preference. 190 total people were
surveyed.
Pro-tax plan
Against tax plan
Republican
70
30
Democrat
10
80
Find the probability that:
a)
person was republican and against tax plan
p(Republican and Against)
= 30/190
b)
person was pro-plan, given republican
p(pro-plan / Republican)
= 70 / (70 + 30)
=70/100
PSY1110: Module 7 Course Notes
=0.1579
=0.70
Page 5
c)
d)
person was democratic, given against plan
p(Democratic / against)
= 80 / (30 + 80)
=80/110
=0.727
the person was pro-plan, given Democratic
p(pro-plan / Democratic)
=10 / (80 + 10)
=10/90
=0.111
PSY1110: Module 7 Course Notes
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Review Questions:
1) Given upcoming NBA draft. What is available:
Point Guard (PG)
Shooting Guard (SG)
Center (C)
Small Forward (SF)
Power Forward (PF)
College Experience (CE)
12
21
9
17
16
No College Experience (NCE)
2
4
12
1
5
Find the following probabilities:
a)
b)
c)
d)
e)
f)
p(PF)
p(C and NCE)
p(CE)
p(SF/CE)
p(not SG)
p(CE/PF)
=
=
=
=
=
=
2) You have an urn containing 10 red marbles, 6 black marbles, and 3 blue marbles.
a) What’s the probability that you draw a red marble, don’t put it back, then draw another red marble?
b) What’s the probability that you draw a black marble, replace it, then draw a blue one?
c) What’s the probability that you draw a blue marble, don’t replace it, then draw a black one?
3) I have 18 cats: 4 black, 2 white, 5 tabby, 2 yellow, 3 gray, and 2 calico.
a)
What’s the probability of me selecting a gray cat?
b)
What’s the probability of me selecting a yellow cat or a black cat?
c)
What’s the probability of me selecting a calico on the first draw and a white cat on the second
draw? Determine with replacement and without replacement.
d)
What’s the probability of me selecting a black or white cat first and a yellow cat on the second
draw? Assume that I have replaced it.
PSY1110: Module 7 Course Notes
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4) Color of my feline friends and sex:
Black
White
Tabby
Male
1
2
2
Female
3
0
3
Yellow
2
0
Gray
1
2
Calico
0
2
a. What’s the probability of me selecting a female cat given that it is gray?
b. What’s the probability of me selecting a female cat given that it is black?
c. What’s the probability of me selecting a tabby cat given that it is a male?
d. What’s the probability of me selecting a white cat and male?
e. What’s the probability of me selecting a gray cat given that it is male?
f. What’s the probability of me selecting a calico cat given that it is female?
g. What’s the probability of me selecting a female given that it is calico?
Answers:
1.
.212 .121
2.
.2632 .0499
3.
.167
4.
.67
.75
.757 .227
.0525
.33
.25
.11
PSY1110: Module 7 Course Notes
.747
.762
.0123 .0131
.125 .20
1.0
.037
Page 8
More Review Questions:
1. When flipping a coin, heads and tails are mutually exclusive because __________.
a. if the coin comes up heads it cannot also come up tails
b. if the coin comes up heads on one toss, it has no influence on whether the coin
comes up heads or tails on the next toss.
c. sampling is with replacement.
d. sampling is without replacement
2. Jake is having a party for all of his friends in his apartment complex. He knows they
all have very different taste, so he stocks up his refrigerator with a large selection. Jake has 12 bottles of Coors
beer, 24 bottles of Molson beer, 24 bottles of Heineken beer, 8 bottles of wine coolers, and 12 bottles of Coke.
2A. Billy wants any beer. What is the probability that the first beverage Billy
randomly grabs is a beer?
2B. Allison wants a Coke. Given that the first bottle grabbed was a Coors, what
is the probability that the second beverage Allison grabs is a Coke?
3. What is the probability of drawing an ace out of a standard deck of 52 cards?
4. What is the probability of drawing a red card of a standard deck of 52 cards?
5. What is the probability of drawing a red ace out of a standard deck of 52 cards?
6. A letter of the English alphabet is chosen at random. Find the probability that the
letter selected…
6A. is a vowel (consider y is a consonant)
6B. is any letter which follows p in the alphabet.
7. There are 105 applicants for a job with the new coffee shop. Some of the applicants have worked at the
coffee shops before and some have not served coffee before. Some of the applicants can work full-time, and
some can only work part-time. The exact breakdown of applicants is as follows…
Coffee Shop
No Coffee Shop
Experience (E)
Experience (not E)
Available
Full-time
(F)
20
12
Available
Part-time
(not F)
42
31
If the order in which the coffee shop manager interviews the applicants is random, E is the event that the
first applicant interviewed has experience working in a coffee shop, and F is the event that the first applicant
interviewed is willing to work full time. Find each of the following probabilities.
7A. P(E) __________
7D. P(E & F) __________
7B. P(F) __________
7E. P(F / E) __________
7C. P(not E) __________
7F. P (not F / not E) __________
PSY1110: Module 7 Course Notes
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Answers:
1. a
2A. 0.75
2B. 0.15
3. 0.0769
4. 0.5
5. 0.0385
6A. 0.1923
6B. 0.3846
7A. 0.5905
7B. 0.3048
7C. 0.4095
7D. 0.1905
7E. 0.3226
7F. 0.7209
PSY1110: Module 7 Course Notes
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