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Chapter 5
Probability in our Daily
Lives
Section 5.1: How can Probability
Quantify Randomness?
Learning Objectives
1. Random Phenomena
2. Law of Large Numbers
3. Probability
4. Independent Trials
5. Finding probabilities
6. Types of Probabilities: Relative Frequency
and Subjective
Learning Objective 1:
Random Phenomena
 For random phenomena, the outcome is
uncertain


In the short-run, the proportion of times that
something happens is highly random
In the long-run, the proportion of times that
something happens becomes very predictable
Probability quantifies long-run randomness
Learning Objective 2:
Law of Large Numbers
 As the number of trials increase, the proportion of
occurrences of any given outcome approaches a
particular number “in the long run”
 For example, as one tosses a die, in the long run 1/6
of the observations will be a 6.
Learning Objective 3:
Probability
 With random phenomena, the probability of a
particular outcome is the proportion of times
that the outcome would occur in a long run of
observations
 Example:

When rolling a die, the outcome of “6” has
probability = 1/6. In other words, the
proportion of times that a 6 would occur in a
long run of observations is 1/6.
Learning Objective 4:
Independent Trials
 Different trials of a random phenomenon are
independent if the outcome of any one trial is
not affected by the outcome of any other trial.
 Example:

If you have 20 flips of a coin in a row that are
“heads”, you are not “due” a “tail” - the
probability of a tail on your next flip is still 1/2.
The trial of flipping a coin is independent of
previous flips.
Learning Objective 5:
How can we find Probabilities?
 Calculate theoretical probabilities based on
assumptions about the random phenomena.
For example, it is often reasonable to assume
that outcomes are equally likely such as
when flipping a coin, or a rolling a die.
 Observe many trials of the random
phenomenon and use the sample proportion
of the number of times the outcome occurs as
its probability. This is merely an estimate of
the actual probability.
Learning Objective 6:
Types of Probability: Relative Frequency vs.
Subjective
 The relative frequency definition of probability is the
long run proportion of times that the outcome occurs
in a very large number of trials - not always
helpful/possible.
 When a long run of trials is not feasible, you must rely
on subjective information. In this case, the subjective
definition of the probability of an outcome is your
degree of belief that the outcome will occur based on
the information available.

Bayesian statistics is a branch of statistics that uses
subjective probability as its foundation
Chapter 5: Probability in
Our Daily Lives
Section 5.2: How Can We Find Probabilities?
Learning Objectives
1. Sample Space
2. Event
3. Probabilities for a sample space
4. Probability of an event
5. Basic rules for finding probabilities about a
pair of events
Learning Objectives
6. Probability of the union of two events
7. Probability of the intersection of two events
Learning Objective 1:
Sample Space
 For a random phenomenon, the sample
space is the set of all possible outcomes.
Learning Objective 2:
Event
 An event is a subset of the sample space
 An event corresponds to a particular outcome
or a group of possible outcomes.
 For example;


Event A = student answers all 3 questions
correctly = (CCC)
Event B = student passes (at least 2 correct) =
(CCI, CIC, ICC, CCC)
Learning Objective 3:
Probabilities for a sample space
Each outcome in a sample space has a
probability
 The probability of each individual outcome is
between 0 and 1
 The total of all the individual probabilities
equals 1.
Learning Objective 4:
Probability of an Event
 The probability of an event A, denoted by
P(A), is obtained by adding the probabilities
of the individual outcomes in the event.
 When all the possible outcomes are equally
likely:
number of outcomes in event A
P( A) 
number of outcomes in the sample space
Learning Objective 4:
Example: What are the Chances of being Audited?
 What is the sample space for selecting a
taxpayer?
{(under $25,000, Yes), (under $25,000, No),
($25,000 - $49,000, Yes) …}
Learning Objective 4:
Example: What are the Chances of being Audited?
For a randomly selected taxpayer in 2002,
 What is the probability of an audit?

310/80200=0.004
 What is the probability of an income of
$100,000 or more?

10700/80200=0.133
 What income level has the greatest
probability of being audited?

$100,000 or more = 80/10700= 0.007
Learning Objective 5:
Basic rules for finding probabilities about a pair
of events
 Some events are expressed as the outcomes
that



Are not in some other event (complement of
the event)
Are in one event and in another event
(intersection of two events)
Are in one event or in another event (union of
two events)
Learning Objective 5:
Complement of an event
 The complement of an event A consists of all
outcomes in the sample space that are not in
A.
 The probabilities of A and of Ac add to 1
 P(Ac) = 1 – P(A)
Learning Objective 5:
Disjoint events
 Two events, A and B, are disjoint if they do
not have any common outcomes
Learning Objective 5:
Intersection of two events
 The intersection of A and B consists of
outcomes that are in both A and B
Learning Objective 5:
Union of two events
 The union of A and B consists of outcomes
that are in A or B or in both A and B.
Learning Objective 6:
Probability of the Union of Two Events
Addition Rule:
For the union of two events,
P(A or B) = P(A) + P(B) – P(A and B)
If the events are disjoint, P(A and B) = 0, so
P(A or B) = P(A) + P(B)
Learning Objective 6:
Example
 80.2 million tax payers (80,200 thousand)
 Event A = being audited
 Event B = income greater than $100,000
 P(A and B) = 80/80200=.001
Learning Objective 7:
Probability of the Intersection of Two Events
Multiplication Rule:
For the intersection of two independent events,
A and B,
P(A and B) = P(A) x P(B)
Learning Objective 7:
Example
 What is the probability of getting 3 questions
correct by guessing?
 Probability of guessing correctly is .2
What is the probability that a student
answers at least 2 questions correctly?
P(CCC) + P(CCI) + P(CIC) + P(ICC) =
0.008 + 3(0.032) = 0.104
Learning Objective 7:
Assuming independence
 Don’t assume that events are independent
unless you have given this assumption
careful thought and it seems plausible.
Learning Objective 7:
Events Often Are Not Independent
 Example: A Pop Quiz with 2 Multiple Choice
Questions

Data giving the proportions for the actual
responses of students in a class
Outcome: II
IC
Probability: 0.26 0.11
CI
CC
0.05
0.58
Learning Objective 7:
Events Often Are Not Independent
 Define the events A and B as follows:
A: {first question is answered correctly}
 B: {second question is answered correctly}
 P(A) = P{(CI), (CC)} = 0.05 + 0.58 = 0.63
 P(B) = P{(IC), (CC)} = 0.11 + 0.58 = 0.69
 P(A and B) = P{(CC)} = 0.58

 If A and B were independent,
P(A and B) = P(A) x P(B) = 0.63 x 0.69 = 0.43
Thus, in this case, A and B are not independent!
Chapter 5
Probability in Our Daily
Lives
Section 5.3: Conditional Probability:
What’s the Probability of A, Given B?
Learning Objectives
1. Conditional probability
2. Multiplication rule for finding P(A and B)
3. Independent events defined using
conditional probability
Learning Objective 1:
Conditional Probability
 For events A and B, the conditional probability of
event A, given that event B has occurred, is:
P( A and B)
P( A | B) 
P( B)
 P(A|B) is read as “the probability of event A, given
event B.” The vertical slash represents the word
“given”. Of the times that B occurs, P(A|B) is the
proportion of times that A also occurs
Learning Objective 1:
Conditional Probability
Learning Objective 1:
Example 1
Learning Objective 1:
Example 1
Learning Objective 1:
Example 1
 What was the probability of being audited,
given that the income was ≥ $100,000?


Event A: Taxpayer is audited
Event B: Taxpayer’s income ≥ $100,000
P(A and B) 0.0010
P(A | B) 

 0.007
P(B)
0.1334
Learning Objective 1:
Example 1
 What is the probability of being audited given that the
income level is < $25,000
 Let A =Event being Audited
 Let B = Income < $25,000
P( A and B)
P( A | B) 
P( B)
 P(A and B) = .0011
 P(B)=.1758
 .0011/.1758=.0063
Learning Objective 1:
Example 2
 A study of 5282 women aged 35 or over
analyzed the Triple Blood Test to test its
accuracy
Learning Objective 1:
Example 2
 A positive test result states that the
condition is present
 A negative test result states that the
condition is not present
 False Positive: Test states the condition
is present, but it is actually absent
 False Negative: Test states the condition
is absent, but it is actually present
Learning Objective 1:
Example 2
 Assuming the sample is representative of the
population, find the estimated probability of
a positive test for a randomly chosen
pregnant woman 35 years or older
 P(POS) = 1355/5282 = 0.257
Learning Objective 1:
Example 2
 Given that the diagnostic test result is positive,
find the estimated probability that Down
syndrome truly is present
P(D and POS)
48 / 5282
P(D | POS) 


P(POS)
1355 / 5282
0.009
 0.035
0.257
 Summary: Of the women who tested positive,
fewer than 4% actually had fetuses with Down
syndrome
Learning Objective 2:
Multiplication Rule for Finding P(A and B)
 For events A and B, the probability that A
and B both occur equals:


P(A and B) = P(A|B) x P(B)
also
P(A and B) = P(B|A) x P(A)
Learning Objective 2:
Example
 Roger Federer – 2006 men’s champion in
the Wimbledon tennis tournament



He made 56% of his first serves
He faulted on the first serve 44% of the
time
Given that he made a fault with his first
serve, he made a fault on his second serve
only 2% of the time
Learning Objective 2:
Example
 Assuming these are typical of his serving
performance, when he serves, what is the
probability that he makes a double fault?
 P(F1) = 0.44
 P(F2|F1) = 0.02
 P(F1 and F2) = P(F2|F1) x P(F1)
= 0.02 x 0.44 = 0.009
Learning Objective 3:
Independent Events Defined Using Conditional
Probabilities
 Two events A and B are independent if the
probability that one occurs is not affected
by whether or not the other event occurs
 Events A and B are independent if:
P(A|B) = P(A), or equivalently, P(B|A) = P(B)
 If events A and B are independent,
P(A and B) = P(A) x P(B)
Learning Objective 3:
Checking for Independence
 To determine whether events A and B are
independent:



Is P(A|B) = P(A)?
Is P(B|A) = P(B)?
Is P(A and B) = P(A) x P(B)?
 If any of these is true, the others are also true and
the events A and B are independent
Learning Objective 3:
Example
 The diagnostic blood test for Down syndrome:
POS = positive result
NEG = negative result
D = Down Syndrome
DC = Unaffected
Status
POS
NEG
Total
D
0.009
0.001
0.010
Dc
0.247
0.742
0.990
Total
0.257
0.743
1.000
Learning Objective 3:
Example: Checking for Independent
 Are the events POS and D independent or
dependent? Is P(POS|D) = P(POS)?
 P(POS|D) =P(POS and D)/P(D)
= 0.009/0.010 = 0.90
 P(POS) = 0.257
 The events POS and D are dependent
Chapter 5: Probability in
Our Daily Lives
Section 5.4: Applying the Probability Rules
Learning Objectives
1. Is a “Coincidence” Truly an Unusual Event?
2. Probability Model
3. Probabilities and Diagnostic Testing
4. Simulation
Learning Objective 1:
Is a “Coincidence” Truly an Unusual Event?
 The law of very large numbers states that
if something has a very large number of
opportunities to happen, occasionally it
will happen, even if it seems highly
unusual
Learning Objective 1:
Example: Is a Matching Birthday Surprising?
 What is the probability that at least two
students in a group of 25 students have
the same birthday?
Learning Objective 1:
Example: Is a Matching Birthday Surprising?
 P(at least one match) = 1 – P(no matches)
Learning Objective 1:
Example: Is a Matching Birthday Surprising?
 P(no matches) = P(students 1 and 2 and 3
…and 25 have different birthdays)
Learning Objective 1:
Example: Is a Matching Birthday Surprising?
 P(no matches) =
(365/365) x (364/365) x (363/365) x …
x (341/365)
 P(no matches) = 0.43
Learning Objective 1:
Example: Is a Matching Birthday Surprising?
 P(at least one match) =
1 – P(no matches) = 1 – 0.43 = 0.57
Not so surprising when you consider that there
are 300 pairs of students who can share the
same birthday!
Learning Objective 2:
Probability Model
 We’ve dealt with finding probabilities in many
idealized situations
 In practice, it’s difficult to tell when outcomes
are equally likely or events are independent
 In most cases, we must specify a probability
model that approximates reality
Learning Objective 2:
Probability Model
 A probability model specifies the possible
outcomes for a sample space and provides
assumptions on which the probability
calculations for events composed of these
outcomes are based
 Probability models merely approximate reality
Learning Objective 2:
Example: Probability Model
 Out of the first 113 space shuttle missions
there were two failures
 What is the probability of at least one failure
in a total of 100 missions?

P(at least 1 failure)=1-P(0 failures)
=1-P(S1 and S2 and S3 … and S100)
=1-P(S1)xP(S2)x…xP(S100)
=1-[P(S)]100=1-[0.971]100=0.947
Learning Objective 2:
Example: Probability Model
 This answer relies on the assumptions of


Same success probability on each flight
Independence
These assumptions are suspect since other
variables (temperature at launch, crew
experience, age of craft, etc.) could affect the
probability
Learning Objective 3:
Probabilities and Diagnostic Testing
 Sensitivity = P(POS|S)
 Specificity = P(NEG|SC)
Learning Objective 3:
Example: Probabilities and Diagnostic Testing
Random Drug Testing of Air Traffic
Controllers
 Sensitivity of test = 0.96
 Specificity of test = 0.93
 Probability of drug use at a given time ≈
0.007 (prevalence of the drug)
Learning Objective 3:
Example: Probabilities and Diagnostic Testing
What is the probability of a positive test result?
P(POS)=P(S and POS)+P(SC and POS)
 P(S and POS)=P(S)P(POS|S)
= 0.007x0.96=0.0067

P(SC and POS)=P(SC)P(POS|SC)
= 0.993x0.07=0.0695

P(POS)=.0067+.0695=0.0762
Even though the prevalence is < 1%, there is an almost
8% chance of the test suggesting drug use!
Learning Objective 4:
Simulation
Some probabilities are very difficult to find
with ordinary reasoning. In such cases,
we can approximate an answer by
simulation.
Learning Objective 4:
Simulation
Carrying out a Simulation:




Identify the random phenomenon to be
simulated
Describe how to simulate observations
Carry out the simulation many times (at least
1000 times)
Summarize results and state the conclusion