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Chapter 5
Modeling
Variation with
Probability
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Slide 1
Chapter 5 Topics
• Use empirical and theoretical probability
models
• Apply basic probability rules
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Slide 2
Section 5.1
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WHAT IS RANDOMNESS?
• Use a Random Number Table or Technology to
Simulate Randomness
• Distinguish Between Empirical and Theoretical
Probabilities
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Slide 3
What Is Randomness?
• Having no predictable pattern
• People are not good at identifying truly
random samples or random experiments.
• Usually a computer or some other
randomizing device, such as a random number
table, is used to simulate randomness.
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Slide 4
Example: Simulating Randomness
Using a Random Number Table
Use Row 30 from the random number table to
simulate 10 coin tosses. Let odd numbers = H
and even numbers = T. What is the longest
streak of H or T in your data?
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Slide 5
Example: Simulating Coin Tosses Using
a Random Number Table
HTHTH HHTHT THHHH TTTHT
Longest streak: 4 heads in a row
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Slide 6
Probability
• Used to measure how often random events
occur
• When tossing a coin, the probability of a head
is ½ or 50%. This means that the coin will land
on heads about 50% of the time.
• Two types:
1. Theoretical
2. Empirical
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Theoretical Probabilities
• Long-run relative frequencies
• The relative frequency at which an event
occurs after infinitely many repetitions
Example: If we were to flip a coin infinitely
many times, exactly 50% of the flips would be
heads.
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Slide 8
Empirical Probabilities
• Relative frequencies based on an experiment
or on observation of a real-life process
Example: I toss a coin 10 times and get 4 heads.
The empirical probability of getting heads is
4/10 = 0.4, or 40%.
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Theoretical vs. Empirical Probabilities
• Theoretical probabilities are always the same
value.
Example: The theoretical probability of getting a heads when
tossing a coin is always 0.50 or 50%.
• Empirical probabilities change with every
experiment.
Example: If I toss a coin 10 times and get 7 heads, the empirical
probability of heads = 0.70 or 70%. If I toss a coin 10 times and
get 3 heads, the empirical probability of heads = 0.30 or 30%.
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Slide 10
Why Two Probability Models?
• Theoretical probabilities may be difficult to
compute – empirical probabilities can help us
estimate theoretical probabilities.
• We may not trust the theoretical probability
model (for example, the model may be based
on faulty assumptions) – empirical
probabilities can help us verify or refute a
theoretical value.
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Slide 11
Simulations
• Experiments used to produce empirical
probabilities
• In a previous example, we used a random
number table to simulate tossing a coin 10
times.
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Slide 12
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Section 5.2
FINDING THEORETICAL
PROBABILITIES
• Find Theoretical Probabilities of Equally Likely Events
• Find the Probability of the Complement of an Event
• Use Basic Probability Rules
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Slide 13
Probabilities
• Always between 0 and 1 (including 0 and 1)
Written in symbols: 0 ≤ P(A) ≤ 1
• Can be expressed as fractions, decimals, or
percents
• An event has a probability of 0 only if it can
never happen.
• An event has a probability of 1 only if it is
certain to happen.
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Slide 14
Probability Notation
Events are usually represented with uppercase
letters: A, B, C, etc. The probability of event A
occurring is written P(A).
Example: Toss a coin. Let A represent the event
“the coin lands on heads.”
P(A) = ½ or 0.50 or 50%
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The Complement of an Event: AC
The complement refers to the event “not”
occurring. The complement of event A is written
Ac .
Example: Toss a coin. A = coin lands on heads,
Ac = coin does NOT land on heads.
Example: Wait for a bus and note its arrival time.
A = bus arrives on time, Ac = bus does NOT arrive on
time.
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The Probability of an Event
and Its Complement
P(A does NOT occur) = 1 – P(A does occur)
In symbols, P(Ac) = 1 – P(A)
Examples: Let event A represent the event “the
candidate wins an election.” Suppose P(A)=0.70.
Then Ac represent the event “the candidate does
NOT win the election” and
P(Ac) = 1 – P(A) = 1 – 0.70 = 0.30
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Slide 17
Sample Space and Events for Equally
Likely Outcomes
Sample space The set of all possible equally
likely outcomes of an experiment.
Event Any collection of outcomes in the
sample space.
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Slide 18
Example: Sample Space
Experiment: Roll a fair die one time.
Sample space: S = 1, 2, 3, 4, 5, 6
Example: Event = “roll an even number”
Outcomes: 2, 4, 6
Example: Event = “roll a number less than 3”
Outcomes: 1, 2
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Slide 19
Using a Sample Space to find a
Theoretical Probability
For equally likely events:
P(A) = number of outcomes in A
number of all possible outcomes
Example: Roll a die. Let A represent the event “get a
number less than 3.” Find P(A).
Sample space: 1, 2, 3, 4, 5, 6 6 outcomes
A: 1, 2 2 outcomes
P(A) =
2
6
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Slide 20
Using a Sample Space to find a
Theoretical Probability
A family has 2 children. Find the probability that
both children are girls.
Let A represent the event that both are girls.
Sample space: BB, BG, GB, GG 4 outcomes
Event A: GG 1 outcome
1
P(A) = = .25 = 25%
4
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Slide 21
Combining Events with “And”
An event belonging to “A and B” must belong to
both A and B.
Example: If event A represents “wearing a hat”
and event B represents “raising a hand” then
someone in A and B is wearing a hat and raising
a hand.
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Slide 22
Combining Events Using “And”
A person is selected at random from this group. Find
the probability that he/she is wearing a hat AND raising
his/her hand. Number in sample space = 6, number
raising a hand and wearing a hat = 2 (Maria and David),
P(A and B) = 2/6.
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Slide 23
Combining Events Using “Or”
An event belonging to “A or B” must belong to A
or B or both.
Example: If event A represents “wearing a hat”
and event B represents “raising a hand” then
someone in A or B is wearing a hat or raising a
hand or both.
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Slide 24
Combining Events Using “Or”
A person is selected at random from this group. Find
the probability that he/she is wearing a hat OR raising
his/her hand. Number in sample space = 6, number
raising a hand or wearing a hat or both (Rena, Maria,
David) = 3, P(A or B) = 3/6
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Slide 25
Finding Probabilities
from Two-Way Tables
In the 2012 General Social Survey (GSS) people were
asked about their happiness and were also asked
whether they agreed with the following statement: “In
a marriage, the husband should work and the wife
should take care of the home.” The following table
summarizes the data collected:
Agree
Happy
242
Don’t
Know
65
Disagree
Total
684
991
Unhappy
45
30
80
155
Total
287
95
764
1146
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Finding Probabilities
from Two-Way Tables
Agree
Happy
Unhappy
Total
242
45
287
Don’t
Know
65
30
95
Disagree
Total
684
80
764
991
155
1146
Suppose a person is randomly selected from this
group. Find:
a. P(happy and agree)
b. P(happy or agree)
c. P(Agree or Don’t Know)
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Slide 27
Finding Probabilities
from Two-Way Tables
Agree
Happy
Unhappy
Total
242
45
287
Don’t
Know
65
30
95
Disagree
Total
684
80
764
991
155
1146
Suppose a person is randomly selected from this
group. Find:
a. P(happy and agree) = 242/1146 = 0.211
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Finding Probabilities
from Two-Way Tables
Happy
Unhappy
Total
Agree
Don’t
Know
Disagree
Total
242
45
287
65
30
95
684
80
764
991
155
1146
Suppose a person is randomly selected from this
group. Find:
b. P(happy or agree)
The total number of people who are happy or agree is:
242 + 65 + 684 + 45 = 1036
P(happy or agree) = 1036/1146 = 0.904
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Slide 29
Finding Probabilities
from Two-Way Tables
Agree
Happy
Unhappy
242
45
Don’t
Know
65
30
Total
287
95
Disagree
Total
684
80
991
155
764
1146
c. P(Agree or Don’t Know)
The total number of people who Agree or Don’t Know
is 242 + 45 + 65 + 30 = 382
P(Agree or Don’t Know) = 382/1146 = 0.333 or 33.3%
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Slide 30
Mutually Exclusive Events
Agree
Happy
Unhappy
242
45
Don’t
Know
65
30
Total
287
95
Disagree
Total
684
80
991
155
764
1146
When counting the number of people who Agree or
Don’t Know, no person was in BOTH categories. We say
the events “Agree” and “Don’t Know” are MUTUALLY
EXCLUSIVE EVENTS.
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Slide 31
Mutually Exclusive Events
Agree
Happy
Unhappy
242
45
Don’t
Know
65
30
Total
287
95
Disagree
Total
684
80
991
155
764
1146
When counting the number of people who are happy
or agree, notice that there is a group of people (242)
who appear in both categories at the same time –
those who are happy AND agree. The events “Happy”
and “Agree” are NOT mutually exclusive.
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Slide 32
Probability Rule: “OR”
P(A or B) = P(A) + P(B) – P(A AND B)
NOTE: If A and B are mutually exclusive, then
they cannot happen at the same time, so P(A
AND B) = 0.
For MUTUALLY EXCLUSIVE EVENTS:
P(A or B) = P(A) + P(B)
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Slide 33
Example: Probabilities with “OR”
Experiment: Roll a fair six-sided die. Find:
a. P(rolling an odd number or a number greater
than 3)
b. P(rolling a number less than 3 or rolling a 6)
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Slide 34
Example: Probabilities with “OR”
Experiment: Roll a fair six-sided die. Find:
a. P(rolling an odd number or a number greater
than 3)
The outcomes for event “rolling an odd number” are 1, 3, 5
and the probability of this event is 3/6. The outcomes for
event “a number greater than 3” are 4, 5, 6 and the
probability of this event is 3/6. Notice these are NOT
mutually exclusive since both events contain the outcome
“5”.
P(rolling an odd number or a number greater than 3) =
3/6 + 3/6 – 1/6 = 5/6.
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Slide 35
Example: Probabilities with “OR”
Experiment: Roll a fair six-sided die. Find:
b. P(rolling a number less than 3 or rolling a 6)
The outcomes for event “rolling a number less than 3”
are 1 and 2, and the probability of this event is 2/6.
The outcome for the event “rolling a 6” is simply 6, and
the probability of this event is 1/6. Notice that these
are mutually exclusive events.
P(rolling a number less than 3 or rolling a 6) = 2/6 + 1/6
= 3/6 or ½.
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Slide 36
Section 5.3
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ASSOCIATIONS IN CATEGORICAL
VARIABLES
• Find Conditional Probabilities
• Determine if Events are Dependent or
Independent
• Use Probabilities to Determine if an Association
May Exist Between Categorical Variables
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Slide 37
Associations Between
Categorical Variables
This table shows marital status and educational level
for a random sample.
Question: Is there an association between marital status and
educational level? In other words, are the proportions of
married people different for various levels of education? For
example, are college-educated more (or less) likely to be married
than HS-educated?
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Slide 38
Conditional Probabilities
Probabilities where we focus on just one group
and imagine taking a random sample from that
group alone are called conditional probabilities.
Example: P(a person is married given that the person is
college-educated)
Example: P(a person is single given that the person’s
highest educational level is HS)
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Slide 39
“Given That” vs. “And”
P(married and college educated) would use the number
in the intersection of married and college.
P(married given that they are college educated) would
focus solely on the number of college educated and
count the number in that group who are married.
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Slide 40
Example: “And”
Ed. Level Single
Married Divorced Widow/er Total
Less HS
HS
College
or +
17
68
27
70
240
98
10
59
15
28
30
3
125
397
143
Total
112
408
84
61
665
P(Married and College-Educated) = 98/665 = .147 or 14.7%
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Slide 41
Example: “Given that”
Ed. Level
Single
Married
Divorced
Widow/er
Total
Less HS
17
70
10
28
125
HS
68
240
59
30
397
College or +
27
98
15
3
143
Total
112
408
84
61
665
To find P(Married, given that s/he is college-educated),
we only focus on the college-educated group for our
total and from this group, find the number who are
married.
P(Married, given that s/he is college-educated) =
98/143 = .685 or 68.5%.
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Slide 42
Conditional Probability: Notation
To write P(married, given that s/he is collegeeducated) we write:
P(married | college-educated)
In general, P(A|B) means find the probability
that event A occurs given that event B has
occurred.
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Slide 43
Calculating Conditional Probabilities
One can calculate conditional probabilities from tables
by isolating the group from which you are sampling as
we did in the previous example.
Conditional probabilities can also be calculated by using
this formula (helpful in cases where you do not have
complete information):
P(A and B)
P(A| B)=
P(B)
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Slide 44
Example: Isolating the Sampling Group
Ed. Level
Single
Married
Divorced
Widow/er
Total
Less HS
17
70
10
28
125
HS
68
240
59
30
397
College or +
27
98
15
3
143
Total
112
408
84
61
665
Find P(married|less HS).
By isolating the sampling group, the total of less HS is
125. Of these, 70 are married, so P(married|less HS) =
70/125 = 0.56
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Slide 45
Example: Using the Formula
Ed. Level
Single
Married
Divorced
Widow/er
Total
Less HS
17
70
10
28
125
HS
68
240
59
30
397
College or +
27
98
15
3
143
Total
112
408
84
61
665
Find P(married|less HS).
Using the formula: P(married AND less HS) = 70/665,
P(less HS)=125/665, so P(married|less HS) =
(70/665)/(125/665) = 70/125 = 0.56.
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Slide 46
Independent Events
Variables or events that are not associated are called
INDEPENDENT EVENTS.
Two events are independent if knowledge that one has
happened tells you nothing about whether or not the
other event has happened.
In symbols: A and B are independent events means
P(A|B) = P(A)
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Slide 47
Example: Independent Events
Suppose a card is drawn from a standard deck of
playing cards. Are the events “the card is a club” and
“the card is black” independent?
P(card is a club) = 13/52 = 1/4
P(card is black) = 26/52 = ½
P(card is a club|card is black) = 13/26 = ½
(Note: There are 26 black cards, of which 13 are clubs.)
P(card is club) ≠ P(card is club|card is black) so the
events are NOT independent. The events are
associated.
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Slide 48
Example
Ed. Level
Single
Married
Divorced
Widow/er
Total
Less HS
17
70
10
28
125
HS
68
240
59
30
397
College or +
27
98
15
3
143
Total
112
408
84
61
665
Are the events “person selected has a HS education”
and “person selected is divorced” independent?
P(HS|divorced) = 59/84 = 0.702
P(HS)=397/665 = 0.597
The probabilities are not equal so the events are not
independent. The events are associated.
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Slide 49
Probability and “AND”
The Multiplication Rule:
For INDEPENDENT events,
P(A AND B) = P(A) P(B).
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Slide 50
Example: Gender of Babies
Suppose 49% of babies born in the US are girls.
A couple has 2 children. Find P(both are girls).
P(first is a girl and second is a girl) =
P(first is a girl) P(second is a girl) =
0.49 x 0.49 = 0.24
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Slide 51
Example: Gender of Babies
A couple has 2 children. Find P(the first is a boy
and the second is a girl).
P(first is a boy and second is a girl) =
P(first is a boy) P(second is a girl) =
0.51 x 0.49 = 0.2499
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Slide 52
Example
According to a Wall Street Journal survey, 70% of
consumers prefer to shop online rather than in person
at their favorite retailer. Suppose three consumers are
randomly selected with replacement from the
population of consumers.
1. What is the probability that all three prefer to shop
online?
2. What is the probability that none prefer to shop
online?
3. What is the probability that at least one prefers to
shop online?
Source: www.wsj.com
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Slide 53
Example
NOTE: These are independent events because one consumer’s
answer will not affect the probability of the next consumer’s
answer, so we can use the Multiplication Rule.
1. P(all three prefer to shop online) = P(first prefers
online AND second prefers online AND third prefers
online)
= 0.70 x 0.70 x 0.70 = 0.343
2. P(none prefer online) = P(first does not prefer online
AND second does not prefer online AND third does
not prefer online)
= 0.30 x 0.30 x 0.30 = 0.027
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Slide 54
Example
3. P(at least one prefers online) = P(one prefers online
OR two prefer online OR three prefer online). We
could calculate the probability of each of these
events and add them together, but it is easier to
note that “at least one is satisfied” is the
complement of “none is satisfied” since is includes
all categories except “none.”
So P(at least one prefers online) = 1 – P(none prefer
online)
= 1 – 0.027 = 0.973
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Slide 55
Section 5.4
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FINDING EMPIRICAL PROBABILITIES
• Use Simulations to Find Empirical Probabilities
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Slide 56
Empirical Probabilities
• Based on observations of real-life events
• Examples:
– Baseball player’s batting average
– Percentage of times a bus arrives late
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Slide 57
Empirical Probabilities
When we can’t find data or when a situation is
too complex for us to find the empirical
probability of a random event, we can
sometimes simulate the situation to generate
data needed to find the empirical probability.
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Steps for a Simulation
1. Identify the random action and the probability of a
successful outcome.
2. Determine how to simulate this random action
(for example: technology, random number table).
3. Determine the event you’re interested in.
4. Explain how you will simulate one trial.
5. Carry out a trial, record whether or not the event of
interest occurred.
6. Repeat the trial many times (at least 100) and count
the number of times your event occurred.
7. Use the data to find the empirical probability.
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Slide 59
Example: Simulation for P(3 Heads)
When Tossing a Coin 3 Times
1. Identify the random action and the probability of a
successful outcome.
Random action is the outcome of a single coin toss.
P(head) = 0.50
2. Determine how to simulate this random action
(for example: technology, random number table).
Use a random number table.
Let even digits = “tails” and odd digits = “heads.”
3. Determine the event you’re interested in.
Interested in whether we get 3 Heads in 3 tosses.
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Slide 60
Example: Coin Toss Simulation
4. Explain how you will simulate one trial.
Read off 3 digits from a row in the table; each digit
represents a toss. Record T for even numbers and H for
odd numbers.
5. Carry out a trial, record whether or not the event of
interest occurred.
If 3 heads occur in 3 tosses, record “yes”; otherwise record
“no.”
6. Repeat the trial many times (at least 100) and count
the number of times your event occurred.
7. Use the data to find the empirical probability.
Empirical probability = #yes/total number of trials
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2014Pearson
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Slide 61
Example: Coin Toss Simulation
Starting with line 01:
210 = THT = NO (did not get 3 heads)
333 = HHH = YES (did get 3 heads)
252 = THT = NO (did not get 3 heads)
Repeat at least 100 times
Copyright
Copyright©©2017,
2017,2014
2014Pearson
PearsonEducation,
Education,Inc.
Inc.
Slide 62
Law of Large Numbers
If an experiment with a random outcome is
repeated a large number of times, the empirical
probability of an event is likely to be close to the
true probability. The larger the number of
repetitions, the closer together these
probabilities are likely to be.
Copyright
Copyright©©2017,
2017,2014
2014Pearson
PearsonEducation,
Education,Inc.
Inc.
Slide 63