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MODULE 4:
PROBABILITY AND
APPLICATION
Module 4.1 Objectives
Students will be able to:
o know the difference between Random
phenomenon and haphazard phenomenon.
Module 4.1: Probability
Review: Two Types of Statistics
Statistics
Probability is the basis
for inferential statistics
Descriptive
•
•
•
•
Inferential
Collecting
Organizing
Summarizing
Presenting
• Hypothesis
Testing
• Determining
Relationships
• Making
predictions
Definitions of Basic Concepts
Definition
Example
An experiment is a situation involving
chance or probability that leads to a
result
The flipping of a coin
The result of an experiment is the
outcome
Heads or tails.
An event is one or more outcomes of
an experiment
A coin landing on heads in each of 3
trials is an event that contains one
outcome (i.e., HHH). Two heads “HHT,
HTH, THH” is an event which contains
three outcomes.
All possible outcomes in an
experiment is the sample space
In an experiment flipping three coins
(H = Head; T = Tail): {HHH, HHT, HTH,
HTT, THH, THT, TTH, TTT}
o understand the concept of probability as a
measure of how “likely” a specific event is
happened.
o understand why probability is important for
inferential statistics.
Random Phenomenon
Random phenomenon has individual outcomes that are
not completely predictable, but probabilities associated
with the possible outcomes are well-defined.
Example: Flipping a coin
Haphazard phenomenon has individual outcomes
where the probabilities associated with the possible
outcomes are unknown.
Example: Asking someone to choose heads or tails
Example One: Coin Toss
Flipping a coin is a random
phenomena because the
probability of a specific
outcome is well-defined.
We can conduct a coin flipping
experiment to demonstrate this.
This experiment would include
flipping a coin over an over.
Each coin flip is considered a
trial of the experiment.
The outcome of each trial is
well defined. It can be either
heads or tails.
1
Example One: Coin Toss (cont.)
An event is defined as a combination of outcomes from
a random phenomenon that meet a specific criterion.
For example: The event that a coin lands on heads in
each of 3 trials (i.e., HHH)
Why Probability?
Probability is the basis for inferential statistics.
Everything is NOT predictable.
o Predictions to test hypotheses are never exactly correct. Results
are only more or less likely.
Distinguish random vs. haphazard phenomena.
o All science is based on observations and sample statistics that are
more or less affected by random variation.
Probability is important in public health because it helps
us make predictions based on previous research.
o If an individual tests positive for HIV using the latest HIV test, what
is the likelihood (or probability) that they actually are infected?
Trial 1
Trial 2
Trial 3
Coin flipping experiment with 3 trials
What is Probability?
o What is the risk of developing type II diabetes among persons who
are clinically obese?
Overview of Basic Probability
http://www.khanacademy.org/math/probability/v/basic-probability
Probability is a measure of how “likely” a specific event is
happened.
Probability is defined as the number of observations that
satisfy some criterion divided by the total number of
possible observations. Alternatively,
A basic definition of probability is the likelihood of an
event being true divided by the total number of
possibilities. For example, a fair coin with two sides.
Probability of head is P(head)=0.5
Example One: Coin Toss (cont.)
https://www.khanacademy.org/math/trigonometry/prob_comb/prob_combinatorics_precalc/v/coin-flipping-example
Probability Distribution
If we list all of the probabilities for a random phenomena we
have created the probability distribution.
0 ≤ Probability ≤ 1.0
Probability
distribution for
three fair coins
Three
heads
1/8
(0.125)
Two heads
3/8
(0.375)
No head
1/8
(0.125)
One head
3/8
(0.375)
0.125+ 0.375+ 0.375+ 0.125 =1.0
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Basic Rules of Probability
The value of a probability must fall within
the range from 0 to 1.0.
o Probability of an impossible event (empty set
) is 0, P()=0.
o Probability of a certain event (sample space
S) is 1, P(S)=1.
The sum of probabilities associated with
all possible outcomes (events) for a
random phenomenon must equal 1.0.
Example Two: Marble Bag
The marble bag: We have a bag
with 9 red marbles, 2 blue
marbles, and 3 green marbles in
it.
What is the probability of
randomly selecting a non-blue
marble from the bag?
9
3
2
P(Non-blue marble) = 12/14 = 6/7
Watch the video for details to
calculate probability next slide.
Example Two: Marble Bag (cont.)
http://www.khanacademy.org/math/probability/v/probability-1-module-examples?exid=probability_1
Example Three: Playing Cards
Playing cards and Venn Diagrams
o We have an ordinary deck with 4 suits and each suit
has 13 types of cards A, 1, 2,….10, J, K, Q.
− Probability of randomly selecting a Jack : P(Jack)=4/52=0.077
− Probability of randomly selecting a Heart : P(Heart)= 13/52=0.25
− Probability of randomly selecting a Jack and Heart: P(J and
H)=1/52=0.019
− Probability of randomly selecting a Jack or Heart: P(J or
H)=16/52=4/13=0.308
o Watch the video next slide for the probability
calculations.
Example Three: Playing Cards (cont.)
http://www.khanacademy.org/math/probability/v/probability-with-playing-cards-and-venn-diagrams
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MODULE 4:
PROBABILITY AND
APPLICATION
Module 4.2 Objectives
Students will be able to:
o understand complement of an event and complement
rule.
o understand union of events, disjoint events and simple
Module 4.2:
Rules of Probability
9/5/2017
addition rule.
o understand intersection of events, independent events
and simple multiplication rule.
3
Re-Cap of Basic Rules Probability
If E is an event and is a subset of the sample
space, then
o 0 ≤ P(E) ≤1: Probability of an event E is between 0
and 1 inclusive.
o P()=0: Probability of an impossible event (empty set
) is 0.
o P(S)=1: Probability of a certain event (sample space
S) is 1.
The sum of probabilities associated with all
possible outcomes (events) for a random
phenomenon must equal 1.0.
Rules of Probability
Given knowledge or
assumptions about a
random phenomenon (e.g.,
rolling a dice), we can use
the rules of probability to
determine probabilities
associated with events
defined by outcomes of that
phenomenon.
Complement of an Event
Probability for Complement of Event:
Complement Rule
For any event A, the complement of A is the
For any event A, the probability that A does NOT
event that A will NOT occur (written as Ac)
Venn diagram:
Examples
o If you flip a coin and get a
A
Ac
head, then the compliment is
tail (A=Head, Ac=Tail).
o A patient who is randomly
assigned to one of two groups
“treatment group” or “control
group” (A= treatment group,
Ac=control group).
occur (Ac) is 1 minus the probability that A does
occur. This is called the complement rule.
The probability of Ac is equal to:
P(Ac) = 1 – P(A)
Example: Flipping three coins
If “A” denotes “three heads” {HHH}, then “Ac ” is
“less than three heads”
P(Ac)=1-P(A)=1-1/8=7/8
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Probability for Complement of Event
Rules for Two Events
Rules for determining the probability of two
or more events.
o Simple Addition Rule.
o Simple Multiplication Rule.
o General Addition Rule.
o General Multiplication Rule.
Examples: Union of Two Events
Union of Events
The “union” of two events occurs when a single
observation fulfills the criteria for one OR the
other of the events.
Notation: A or B
A
Venn diagram:
Persons who are
African-American or
Hispanic
Patients who are
obese or diabetic
B
Example: A fair die is rolled
A=“getting a number divisible by 2” that has outcome {2,4,6} and
B=“getting a number divisible by 3” that has outcomes {3,6}. Then,
the union of A and B is {2,3,4,6}.
African
American
Hispanic
Obese
Disjoint (Mutually Exclusive) Events
Examples: Disjoint Events
A single observation (individual) cannot fulfill the
Rolling a six-sided die
cannot be both “getting 1
or 2” and “getting 6”
(Disjoint)
criteria for both events
Venn diagram:
A
B
Getting
1 or 2
Getting
6
Diabetic
Flipping a coin cannot
be both “heads” and
“tails” (Disjoint)
Heads
Tails
Determining whether two events are disjoint is
based on knowledge of the process that
produces the events, NOT statistical
knowledge.
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Examples: Not Disjoint Events
Probability for the Union of Two Events:
Simple Addition Rule
A single patient can be both “obese” and
“diabetic” (Not Disjoint).
If the two events are disjoint, the probability
associated with the union of the two events is:
Obese
Diabetic
P(A or B) = P(A) + P(B)
Watch the video in the next slide for an example
A fair die is rolled.
A=“getting a number divisible by 2” that has outcome
{2,4,6} and B=“getting a number divisible by 3” that has
outcomes {3,6}. Two events with sharing a common
outcome {6} are not disjoint .
of the addition rule.
Simple Addition Rule
Simple Addition Rule: Examples
Suppose a fair die is rolled.
A=“getting a number<2” that has outcome {1} and B=“getting a
number>4” that has outcomes {5,6}.
o Two events are disjoint, and union of A and B is {1,5,6}
o P(A or B)=P(<2 or >4)=P(A)+P(B)=1/6+2/6=1/2=0.5
A=“getting a number divisible by 2” that has outcome {2,4,6} and
B=“getting a number divisible by 3” that has outcomes {3,6}.
o Two events are not disjoint, and the union of A and B is {2,3,4,6}
o Probability of this union is 4/6=2/3=0.667 (True probability)
o If the simple addition rule is applied,
P(A or B)=P(A)+P(B)=3/6+2/6=5/6 (Not the true probability)
o Note: This discrepancy is due to “double-counting” of outcome {6}
The Intersection of Two Events
The “intersection” of two events occurs when a
single observation fulfills the criteria for both
A and B
events.
Notation: A and B
Venn diagram:
A
Examples: Intersection of Two Events
A person is Republican
and Hispanic
Hispanic who is a Republican
A patient is obese
and diabetic
Obese and diabetic
B
Example: A fair die is rolled
Republican
Hispanic
Obese
Diabetic
A=“getting a number divisible by 2” that has outcome {2,4,6} and
B=“getting a number divisible by 3” that has outcomes {3,6}. Then, the
intersection of A and B is {6}.
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Independent Events
When the outcome for one event is in no way related to
or influenced by the outcome of the other event.
Determining whether two events are independent is
based on knowledge of the process that produces the
events, NOT statistical knowledge.
Examples
o
o
Dependent & Independent Events
Watch the videos in the next two slides.
The first one is an example of a dependent
event and the second one talks about an
independent event.
Randomly guessing the correct answer to two multiple choice
questions (Independent)
Obese people have a higher risk of becoming diabetic. (Not
Independent)
Example 1: Not Independent Events
http://www.khanacademy.org/math/probability/v/independent-events-1
Probability for the Intersection of Two
Events: Simple Multiplication Rule
Example 2: Independent Events
http://www.khanacademy.org/math/probability/v/independent-events-2
Multiplication Rule
If the two events are independent, the
probability associated with the intersection of
two events is:
P(A and B) = P(A) × P(B)
Watch the video in the next slide for an example
of the multiplication rule.
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Simple Multiplication Rule: Example
On a multiple choice test, problem one has 4
choices and problem two has 3 choices. Each
problem has only one correct answer.
• Correct answer on #1 (A) and correct answer #2
(B) are independent events.
• The probability of randomly guessing the correct
answer to both problems is
P(A and B)= P(A)×P(B)=1/4 × 1/3=1/12=0.0833
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MODULE 4:
PROBABILITY AND
APPLICATION
Module 4.3 Objectives
Students will be able to:
o understand the concept of conditional
probability.
Module 4.3:
Conditional Probability
Conditional Probability
Conditional probability of an event “B” is the
o understand how probability to describe
sensitivity, specificity, positive predictive value,
negative predictive value and prevalence.
Conditional Probability
If events A and B are not independent, then the
probability that the event will occur given the
knowledge that an event “A” has already
occurred. This probability is written by P(B|A).
In the case where events A and B are
independent (where event A has no effect on
the probability of event B), the conditional
probability of event B given event A is simply
the probability of event B, that is P(B).
probability of the intersection of A and B is
defined by
P(A and B) = P(A)P(B|A)
From this definition, the conditional probability
P(B|A) is easily obtained by dividing by P(A):
P(A and B) /P(A)
The next two videos show examples of
conditional probability. The first one is an
example of coin tossing and the second that of
HIV testing.
Example – Tossing a Coin
Example – HIV Testing
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Conditional Probability
Example: Conditional
Diabetic
Dc
D
Obese
probability—proportion
of individuals who were
obese (i.e. were
exposed) had the
diabetic (disease, i.e.
outcome)
Compute:
Application of Conditional Probability
O
P(D O)= 0.40
P(Dc O)= 0.08
P(O)=0.48
Oc
P(D Oc)= 0.04
P(Dc Oc)=0.48
P(Oc)=0.52
P(D)=0.44
Assess test performance.
o Screen for disease by giving test.
o Screening tests are used in clinical practice to assess
the likelihood that a person has a particular medical
condition.
Evaluate the test for accuracy in predicting the
P(Dc)=0.56
disease.
P(D|O)
o Sensitivity and specificity.
o Positive and negative predictive values.
=P(D and O)/P(O)
= 0.4/0.48
Assessing Test Performance
Sensitivity
Data layout for evaluating a screening test:
The test correctly identifies a diseased person as likely to have
the disease
Disease
Disease-free
Probability or proportion of testing positive if you truly have the
Total
Screen +
a
b
a+b
Screen –
c
d
c+d
a+b
b+d
N
Total
disease:
P(screen positive|disease)=a/(a+c)
True disease status
by adding the numbers in the cells horizontally and
vertically.
Test
Total appears at the margins of a table and is calculated
+
–
D
Dc
a
c
b
d
Example – Sensitivity
Sensitivity
Probability or
P(+|D)=P(+D)/P(D)
=0.05/0.06
=0.83
Disease
D
Test
proportion of people
with disease that
test positive
+
–
Dc
P(+D)=0.05
P(+D*)=0.08
P(-D)=0.01
P(-D*)=0.86
P(D)=0.06
P(D*)=0.94
P(+)=0.13
P(-)=0.87
Sensitivity=0.83
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9/5/2017
Specificity
The test will correctly identify a non-diseased person
as un-likely to have the disease
Probability or proportion of testing negative if you truly
do not have the disease:
P(screen negative | disease free)= d/(b+d)
Test
True disease status
+
–
D
Dc
a
c
b
d
Probability or
proportion of people
without disease that
test negative
D
Test
Specificity
P(-|Dc) =P(-Dc)/P(Dc)
= 0.86/0.94
=0.91
Disease
Dc
P(+D)=0.05
P(+Dc)=0.08
P(-D)=0.01
P(-Dc)=0.86
P(D)=0.06
P(Dc)=0.94
+
P(+)=0.13
–
P(-)=0.87
Specificity=0.91
Example – Specificity
Positive Predictive Value (PPV)
Probability or proportion of truly having the
disease if you test positive:
P(disease | test positive) = a/(a+b)
Test
True disease status
+
–
D
Dc
a
c
b
d
Example – Positive Predictive Value
Positive Predictive Value (PPV)
Probability or
P(D|+) =P(+D)/P(+)
= 0.05/0.13
=0.38
Disease
D
Test
proportion that
people who test
positive have the
disease
+
–
Dc
P(+D)=0.05
P(+Dc)=0.08
P(-D)=0.01
P(-Dc)=0.86
P(D)=0.06
P(D c)=0.94
P(+)=0.13
P(-)=0.87
PPV=0.38
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Negative Predictive Value (NPV)
Negative Predictive Value (NPV)
Probability or proportion of not having the
Probability or
Test
True disease status
+
–
D
Dc
a
c
b
d
Disease
Dc
D
Test
proportion that
people who test
negative do not
have the disease
disease if you test negative:
P(disease free | test negative)=d/(c+d)
+
–
P(+D)=0.05
P(+Dc)=0.08
P(-D)=0.01
P(-Dc)=0.86
P(D)=0.06
P(Dc)=0.94
P(Dc|-) =P(-Dc)/P(-)
= 0.86/0.87
=0.99
P(+)=0.13
P(-)=0.87
NPV=0.99
Example – Negative Predictive Value
Prevalence
Proportion of participants with a risk factor or disease at a
particular point in time.
Point Prevalence =
Example
o Prevalence of heart disease among exercisers = 10/40 = 25%
o Prevalence of heart disease among no exercise = 40/60 = 67%
Prevalence
Dependence on Prevalence
D
Test
P(D) =0.06
Depend on prevalence
o Positive predictive value
o Negative predictive value
Disease
proportion of people
with disease
+
–
Dc
P(+D)=0.05
P(+Dc)=0.08
P(-D)=0.01
P(-Dc)=0.86
P(D)=0.06
P(Dc)=0.94
Disease
D
P(+)=0.13
P(-)=0.87
Independent of
prevalence
Test
Probability or
+
–
Dc
P(+D)=0.05
P(+Dc)=0.08
P(-D)=0.01
P(-Dc)=0.86
o Sensitivity
o Specificity
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9/5/2017
Example
Sensitivity: Proportion of
people with disease that test
positive
Diabetic
Dc
D
=.10/.12=.833
Specificity: Proportion
of people without
disease that test
negative
P(-|Dc)= P(-Dc)/P(Dc)=.86/.94
=.805/.88=.91
Test
P(+|D)= P(+D)/P(D) =.05/.06
+
–
.05
.10
.08
.075
.01
..02
.86
.805
.06
.12
.94
.88
.13
.175
.87
.825
1.00
Sensitivity and specificity do
NOT depend on prevalence
5
MODULE 5:
Module 5.1 Objectives
RANDOM VARIABLES AND
PROBABILITY DISTRIBUTIONS
Students will be able to
o understand the concept of random variables
o define and identify discrete variables and
Module 5.1:
Random Variables
Random Variables
continuous variables.
o distinguish discrete vs. continuous variables.
Random Variables
http://www.khanacademy.org/math/probability/v/introduction-to-random-variables (0-3:50 minutes)
A random variable is a quantitative
representation of the outcomes from a random
phenomenon (i.e., whose value is subject to
variations due to randomness).
As opposed to other mathematical variables, a
random variable conceptually does not have a
single, fixed value; rather, it can take a set of
different possible values, each with an
associated probability.
p[X] indicates the probability of getting a
specified value X
Examples: Random Variables
Number of “Heads” out of 4 coin tosses.
Number of people who test positive for HIV in a nationwide sample of 10,000 college students.
Proportion of female subjects in a sample of 100 HIV
infected patients.
Mean of height values from a random sample of 50
university students.
Standard deviation of the 50 height values.
Two Types of Random Variables
Discrete random variable has a finite number of possible values or
an infinite sequence of countable real numbers.
o X = number of “heads” when flipping a coin 4 times.
o X = number of patients who arrive at the USF Student Health
Clinic Monday – Friday.
Continuous random variable takes all values in an interval of real
numbers.
o X = time spent by a physician examining a patient.
o X = time it takes a student to finish an exam.
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Types of Random Variables
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