Basic Probability Theory

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Basic Probability Theory

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Title: Basic Probability Theory

1
BERHANU TAMERU Center for Computational
Epidemiology, Bioinformatics and Risk
Analysis TUSKEGEE UNIVERSITY
Berhanu Tameru, CCEBRA, Tuskegee University
2
General Risk Modelling Approach
3
Risk Analysis
Risk Management
Risk Assessment
Process Initiation
? Science based
? Policy based
Risk Communication
? Interactive exchange of information and
opinions concerning risks
4
Risk Analysis
Process Initiation
Risk Assessment
Risk Management
? Science based
Problem Identification Triad
? Policy based
Problem Management Triad Risk Mitigations
Risk Communication
? Interactive exchange of information and
opinions concerning risks
5
What is Risk Assessment?

Part of the process of Risk Analysis
A scientifically based process composed of 4
steps hazard identification, exposure
assessment, hazard characterization, and risk
characterization
An estimate of the probability of occurrence and
severity of the consequences of exposure to a
pathogen on human health

6
Risk Assessment
7
Terminology - HazardsPotential to cause
disease/injury/harm/damage/loss
8
How is Risk Assessment done?
Risk Assessment consists of answering these
questions

1. What can go wrong that could lead to an
outcome of hazard exposure?
2. How likely is this to happen?
3. If it happens, what consequences are expected?

9
1. What can go wrong that could lead to an
outcome of hazard exposure?

Given a system/process with defined
goals/methodologies
failures of components/procedures can occur
The process of identifying all possible hazards
is called Hazard Analysis.
The process of defining the possible scenarios
that lead to the outcomes or events of interest,
is called Scenario Analysis
The graphic depiction of all the events (failures
of procedures/components) that lead to the
outcome of hazard exposure is called an Event
Tree, Scenario Tree, or Risk Pathway Tree.

10
Terminology Uncertainty

Uncertainty - refers to lack of knowledge about
specific factors, parameters, or models.
Includes
Parameter uncertainty (measurement, sampling, and
systematic errors)
Model uncertainty (due to necessary
simplification of real world processes,
mis-specification of model structure, model
misuse, use of inappropriate surrogate variables)
Scenario uncertainty (descriptive errors,
aggregation errors, errors in professional
judgment, incomplete analysis)
Further measurement or study
Reduces Uncertainty

11
Terminology Variability

Variability - differences attributable to true
heterogeneity, or diversity in population or
parameter. Sources are a result of natural random
processes and stem from environmental/lifestyle/ge
netic differences among humans, animals, plants,
cells, or microbes. Examples of variability
include
Plant, Animal or human physiological variation
(Age, bodyweight, height, blood pressure, heart
rate, drinking water intake rates)
Climatic variability, variation in soil types,
differences in contaminant concentrations
Further measurement or study
Reduces Uncertainty
Better Characterizes Variablity

12
Decision-making under Uncertainty

The purpose of risk assessment is to help
decision-makers make informed decisions about
regulations.
Risk assessments should inform decision-makers
about the level of risk associated with
contemplated or proposed regulatory options.
Risk assessments should also inform
decision-makers about the level/degree of
uncertainty surrounding the risk values presented.

13
Decision-making under Uncertainty

Science based decision making depends on several
analyses (risk, economic, environmental)
The validity, quality objectivity of these
analyses is essential.
The data and assumptions in all analytical
documents used to support decision-making must be
consistent.
Decision Makers need to understand have
confidence in these analyses.
The analyses should stand up to scrutiny

14
Why Probability?
Berhanu Tameru, CCEBRA, Tuskegee University
15
RISK ASSESSMENT MODEL
QUALITATIVE
QUANTITATIVE
Deterministic (Point estimate)
Probabilistic (Range Distribution)
Berhanu Tameru, CCEBRA, Tuskegee University
16
IMPORT RISK ANALYSIS (RA) MODEL
COUNTRY A FROM MULTY COUNTRIES IMPORT RA
COUNTRY A TO COUNTRY B IMPORT RA
GENERIC IMPORT RA
Berhanu Tameru, CCEBRA, Tuskegee University
17
DETERMINSTIC
EXAMPLE IF on average a cow gives 5L milk per
day how much milk can you harvest in 30
days? 5 L per day x 30 days 150 L Both the
input and the output are expressed as single
value
?
OUTPUT
INPUT
RISK ASSESSMENT
POINT VALUES 150 L
POINT VALUES 5 L per day 30 days
Berhanu Tameru, CCEBRA, Tuskegee University
18
Scenario Analysis
What if the cow .. In real
problem there are a lot of factors affecting a
process. Which implies that various states of
phenomenon occur with certain uncertainty and
variability.
Berhanu Tameru, CCEBRA, Tuskegee University
19
PROBABLISTIC
INPUT
OUTPUT
RISK ASSESSMENT
PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION

When uncertainty variability are present, the
chance of any outcome occurring is described by
probability distributions.

Berhanu Tameru, CCEBRA, Tuskegee University
20

Probability is conventionally expressed on a
scale from 0 to 1 a rare event has a probability
close to 0, a very common event has a probability
close to 1.
An outcome is the result of an experiment or
other situation involving uncertainty.
The set of all possible outcomes of a probability
experiment is called a sample space which is
usually denoted by S.
Any subset of the sample space is an event.

Berhanu Tameru, CCEBRA, Tuskegee University
21

Classical Definition
If an event E can occur in r mutually exclusive,
equally likely ways, then the relative frequency
is given by
The number of times the event E occurs ( r )
The total number of times an experiment is
carried out (n).
The probability of an event E, denoted by P(E),
can be thought of as the limiting value of its
relative frequency when the experiment is carried
out many times.

Berhanu Tameru, CCEBRA, Tuskegee University

22

Example 1
Experiment (Posteriori )
Tossing a fair coin 50 times (n 50)
Event E 'heads'
Result 30 heads, 20 tails, so r 30
Relative frequency r/n 30/50 3/5 0.6

Berhanu Tameru, CCEBRA, Tuskegee University
23

Example 2
Classical (Priori )
A herd of 500 animals has 250 infected animals.
The total possible number of outcomes is 500 (n
500)
Event E infected animals
Number of ways event E can occur is 250 (r 250)

Berhanu Tameru, CCEBRA, Tuskegee University
24

Subjective Probability
A subjective probability describes an
individual's personal judgment about how likely a
particular event is to occur. It is not based on
any precise computation but is often a reasonable
assessment by a knowledgeable person.
A person's subjective probability of an event
describes his/her degree of belief in the event.

Berhanu Tameru, CCEBRA, Tuskegee University
25

The empty set an impossible event
P( the empty)0.
S (the sample space) an event that is certain
to occur
Thus
P (the sample space)1
Rules of probability.
1. 1 ? P (x) ? 0 for all values of x.
2. ? p (x) 1

All x
Berhanu Tameru, CCEBRA, Tuskegee University
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Combining probability
Set theory is used to represent relationships
among events.
In general, if A and B are two events in the
sample space S, then
(AU B A union B) 'either A or B occurs or
both occur'

Venn Diagram of A U B
Berhanu Tameru, CCEBRA, Tuskegee University
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(A ? B A intersection B) 'both A and B
occur' Venn Diagram of A ?
B Independent Events Two events are independent
if the occurrence of one of the events gives us
no information about whether or not the other
event will occur that is, the events have no
influence on each other. Thus, P(A ? B)
P(A)P(B)
Berhanu Tameru, CCEBRA, Tuskegee University
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Mortality and product rule example.
Assume that the mortality rate of animals
in a certain herd due to a disease is 15. Thus,
the probability that an animal will die from this
disease is .15 and will survive is .85. If you
buy two animals from this herd what is the
probability that both will die (Assume that they
are independent).
Let A be the event that the first animal dies
and B event that the second animal dies. So we
want to find P(A ? B) .

P(A ? B) P(A)P(B) .15.15
.0225
We can extend this to any possible number of
animals. Let p .15, then for two animals we
have pp p2, for three animals ppp p3 and
in general for n animals we have p n.
Berhanu Tameru, CCEBRA, Tuskegee University
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Additive Rule of Probability
P (A ? B) P (A) P (B) - P (A ? B)
If A and B are mutually exclusive events
(i.e. if one occurs the other can not ), then
P (A ? B)0,
hence
P (A ? B) P (A) P (B)

A ? B Empty set
Berhanu Tameru, CCEBRA, Tuskegee University
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(A the complement of A) 'event A does not
occur'
P(A' ) 1 P(A)

If we wanted to calculate the probability that
there is at least one (i.e. one or more )
infected animal among a group of size n, selected
at random from a herd with prevalence p, then we
proceed as follows
We want to find the complement of non is
infected. The probability that a randomly
selected animal is not infected is 1 p.
From the mortality and product rule example the
probability that none of the n animals is
infected is (1 p) n
The probability that at least one of the n
animals is infected is 1 - (1 p) n

Berhanu Tameru, CCEBRA, Tuskegee University
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Conditional Probability
How the outcome of one event is influenced by
that of another event.
The conditional probability that event A occurs
given that event B occurs can be calculated by
dividing the probability that both A and B occur
by the probability that B occurs that is
P(AB) P (A ? B)
P(B)
P (A ? B) P(B)P(AB) or
P (A ? B) P(A)P(BA)

S
A ? B
B
A
Berhanu Tameru, CCEBRA, Tuskegee University
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Thus, we get Bayes Theorem And in general,
we have Pr( Ai B) Pr(Ai)Pr(BAi)
? Pr(BAj) Pr(Aj)
Prior probability
Likelihood
Posterior probability
All j
Berhanu Tameru, CCEBRA, Tuskegee University
33
Bayes Theorem
A Person has cancer p(A)0.1 (prior)
B Person is smoker p(B)0.5
p(B A)0. 8 (likelihood)
What is p(A B)? (posterior) (i.e. p( The
person has cancer, given that the person is
smoker)
Prior probability
Likelihood
Posterior probability
So p(AB)0.16
Berhanu Tameru, CCEBRA, Tuskegee University
34
We wish to know the probability that an animal
will be infected D , given that it passes, T- ,
a specific veterinary check.
Objective To find Pr( D T-).
Berhanu Tameru, CCEBRA, Tuskegee University
35

From Bayes Theorem
Pr( D T-) P(T-/D) P(D)
P(T-/D) P(D) P(T-/D-) P(D-)
(1-Se)p
(1-Se)p Sp(1-p)
Where p P(D) true prevalence and thus
P(D-)1-p
P(T/D)the sensitivity of the test Se. Hence
P (T-/D)False Negative (1-Se)
P(T-/D-) the specificity of the test Sp. Hence
P(T/D-) False Positive (1 Sp)

Berhanu Tameru, CCEBRA, Tuskegee University
36
Home WorkFrom the Big Bookpage 212 - 214, 218
and 219
Berhanu Tameru, CCEBRA, Tuskegee University
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The Probability Distribution
Berhanu Tameru, CCEBRA, Tuskegee University
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Counting

Experiments?
Coin toss, die roll, DNA sequencing
Basic Principle of Counting
For r number of experiments
First Experiment can have n1 possible outcomes,
second experiment n2, third n3,
Total number of possible outcomes is
n 1 n 2 n 3 n r

39
Factorials

m! m(m-1)(m-2)(m-3)321
5! 54321 120
0! 1
1! 1
100! 1009998321 9 x 10157

40
Permutations

Groups of Ordered arrangements of things
How many 3 letter permutations of the letters a,
b, c are there?
abc, acb, bac, bca, cba, cab ? 6 total

Can use Basic Principle of Counting
321 6

nPr n! .
(n-r)!
n(n -
1)(n -2) ... n - (r - 1)
General Formula

n total number of things
r size of the groups you are taking
r
3!/(3-3)! 6

41
Permutations

For example Find the permutation of 7 from 10.
10P7
10x9x8x7x6x5x4x3x2x1
3x2x1
604880

10!_ (10-7)!
42
Permutations

Remember
In permutations, order is important!
A permutation is a way of determining the number
of elements in the set of outcomes of an
experiment.
Elements in a permutation are selected from a
set without replacement.

43
Permutations

What if some of the things are identical?
How many permutations of the letters
a, a, b, c, c c are there?

Where n1, n2, nr are the number of objects that
are alike
6 letters 2 as, 3 cs and one b Hence
6! / (3!2!) 60
44
Combinations

Groups of things (Order doesnt matter)
How many 3 letter combinations of the letters a,
b, c are there?
1 abc
How many 2 letter combinations of the letters a,
b, c are there?
3 ab, ac, bc
ab ba ac ca bc cb
Order doesnt
matter

45
Combinations
General Formula

n total number of things
k size of the groups you are taking
k

n choose k

Remember In combinations, order means nothing!
Select combination
Order permutation

Random Variable Is a function that assigns one
and only one numerical value to each simple event
of an experiment.
Random variable that can assume a countable
number of values are called Discrete.
Random variable that can assume value
corresponding to any of the points contained in
one or more intervals are called continuous.

Berhanu Tameru, CCEBRA, Tuskegee University
47
Random Variables

Types of variables (defined by state type)
Discrete Labeled , Discrete Numeric
Continuous
Finite or Infinite
If X is a random variable with discrete states
x1, x2, ..., xn

Berhanu Tameru, CCEBRA, Tuskegee University
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Random Variables
Summary Statistics for discrete RVs
Expected Value (mean or arithmetic
average) Variation where Median (state
value where cumulative probability ½)
Berhanu Tameru, CCEBRA, Tuskegee University
49

5. Special Discrete Distributions
In this workshop we study the probabilistic
models (probability distributions) of several
discrete random variables. These random
variables may serve to model a vast number of
applications in many diverse sciences. Among
these distributions are
The Bernoulli distribution.
The binomial distribution.
Poisson distribution and Poisson process.
The geometric distribution.
The negative-binomial (Pascal) distribution.
The hypergeometric distribution.

Bernoulli Trials
The Bernoulli trials process, named after James
Bernoulli. Essentially, the process is the
mathematical abstraction of coin tossing, but
because of its wide applicability, it is usually
stated in terms of a sequence of generic trials
that satisfy the following assumptions
Each trial has two possible outcomes, generically
called success and failure.
The trials are independent. Intuitively, the
outcome of one trial has no influence over the
outcome of another trial.
On each trial, the probability of success is p
and the probability of failure is 1 - p.

Berhanu Tameru, CCEBRA, Tuskegee University
51
Important Distribution
Distribution
52
BINOMIAL DISTRIBUTION
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Binomial Process

A binomial experiment or process is a Bernouli
trial which has the following characteristics
The experiment consists of n identical trials
Each trial results in one of two outcomes (a
success or a failure)
The probability of success on a single trial is
equal to p and remains the same from trial to
trial
The trials are independent (not influenced by the
results of previous tests)
The interest is in x, the number of successes
observed in n trials, for x 0,1,2,..n.

Berhanu Tameru, CCEBRA, Tuskegee University
54
Binomial Distribution

Probability of x successes in n trials of an
experiment where the probability of success in
each trial, p, is constant is equal to the
product of
the number of different ways of getting x
successes in n trials and
the probability of x successes and
the probability of n-x failures

To determine the probability of no successes,
P(X0)
Berhanu Tameru, CCEBRA, Tuskegee University
55
The probability of at least one success is the
same as compliment of no success, i.e. 1
probability of no success.
P(X 1) 1 - P(X0) 1 - (1-p)n
56
Number of Trials needed to achieve s
successes. Sometimes, we know how many successes
s we wish to have, we know the probability p, and
would like to know the number of trials that we
will have to complete in order to achieve the s
successes, assuming we stop once the sth success
has occurred.
Let x be the total number of failures, then the
total number of trials that we will have is (x
s) , and by the (x s -1) trial we achieve the
very last sth successes. Thus the probability of
(s 1) success in (x s -1) trial is given by
the binomial distribution as
P(Xx) (x s -1) C (s -1) (p)s (1-p) x
PNegBin(Xx)
n s x s NegBin(s,p)
Berhanu Tameru, CCEBRA, Tuskegee University
57
The pmf of Negative Binomial
The pmf of the negative binomial r.v. X number
of trials until we observe the r th success, with
parameters r number of successes, is
E(X) r/p, V(X) r(1-p)/p2
58
Binomial Process-for unknown parameters

IF the assumptions of the binomial process can be
satisfied, AND two of the values n,p or x are
known, THEN the third value can be estimated from
the following distributions
Binomial distribution is used to model the number
of successes x
X Binomial( n, p )
Beta distribution is used to model the
probability of success, p
p Beta( x1, n-x1 )
Negative binomial distribution is used to model
the number of trials n, undertaken before x
successes have occurred
n x Negative binomial( x, p )

x Binomial( n, p)
p Beta( x1, n-x1)
n x Negative Binomial( x, p)
Berhanu Tameru, CCEBRA, Tuskegee University
59
Geometric and Negative Binomial Distributions
Let X1, X2, Xn, (independent and identically
distributed) iid-Geom.(p)
Negative binomial random variable represented as
a sum of geometric random variables.
60
HYPERGEOMETRIC DISTRIBUTION
Berhanu Tameru, CCEBRA, Tuskegee University
61
Hypergeometric Distribution

The Hypergeometric distribution returns the
number of items of a particular characteristic
when n items are resampled from a population M
and where D of M items are known to have this
characteristic.

Berhanu Tameru, CCEBRA, Tuskegee University
62
The equation for the hypergeometric distribution
is
P(X x)
where x Defect in the selected sample n
number of sample selected D population
defective N Total population
HYPGEOMDIST is used in sampling without
replacement from a finite population.
Berhanu Tameru, CCEBRA, Tuskegee University
63
Hypergeometric Distribution Example

Suppose 1000 animals are to be imported in a
consignment and 50 are to be tested for a
pathogen. We estimate that if any are infected,
at least 7 will be infected because of
cross-infection by the time the consignment
arrives at port.

Berhanu Tameru, CCEBRA, Tuskegee University
64
Hypergeometric Distribution

M (total population)

1000
D (infected)
7
n (selected)
50
Berhanu Tameru, CCEBRA, Tuskegee University
65
Hypergeometric Distribution Example

If there is any infection in the consignment, the
minimum number of infected animals that will be
tested is estimated as follow
X Hypergeo (50,7,1000)

Berhanu Tameru, CCEBRA, Tuskegee University
66
Hypergeometric Distribution Example
The Hypergeometric (50, 7, 1000)
67
Hypergeometric Distribution

Suppose a company has a stock of 2000 tiles which
is known to contain 70 faulty tiles. The tiles,
unfortunately are all mixed. A customer orders
800 tiles. The probability of getting faulty
tiles (0 to 70) can be calculated from

X Hypergeometric (800,70,2000)
Where X faulty tiles (0-70)
n 800 (customer order)
D 70 (total faulty tiles)
M 2000 (total no. of tiles)

68
Hypergeometric Distribution

The binomial distribution approximate the
hypergeometric distribution when n when the sample size n is small,

69
The Poisson Distribution
70
The Poisson Process

A Poisson process has the following
characteristics
It models the number of events x, that occur in
an interval t, of time or space
It is characterized by one parameter lambda l,
the average number of events per unit interval of
space or time (l 1/b ) where b is the mean
interval between events.
There is a constant and continuous probability of
an event occurring per unit interval
The number of events that occur in any one
interval is independent of the number that occur
in any other interval. It does not matter how far
apart the events are in space or time. For
instance, an event may have only just been
observed or there may have been a considerable
interval between them.
The interval t is measured in either space (per
l,kg,km) or time (per hour/day/year)

71
Poisson Distribution

Average number events that occur per unit of
exposure ?
Random variable X number of events that occur
in t units of exposure.
Prob. Mass Function Pr(x ?,t) (?t)xe-?t
x!
X 0,1,2,
?t 0
Mean events in t units of exposure ?t
Variance ?t

72
Poisson Distribution, cont.
Pr(x ?,t) (?t)xe-?t x!
X 0,1,2, and ?t 0

Outcome is events in amount of exposure.
Events do not have equivalent of failure and
success. Event might be earthquakes in 1 year. No
meaningful definition of non-event.
Medium of exposure is continuous (e.g., time,
volume, etc.), not discrete like trial in
Binomial.
No upper bound to random variable as in binomial.
Note that syntax is different in Excel and _at_Risk.

73
Poisson Distribution, syntax in Excel, _at_Risk

Syntax used in most texts
Pr(x ?,t) (?t)xe-?t
x!
Where ? average number events per unit
exposure, e.g. 3 infestations per year.
b average interval between events 1/ ?
Syntax in Excel POISSON(x, expected value?t,
0)
and _at_Risk Poisson(expected value?t)

74
Poisson Process distributions

Gamma distribution is used to model
a distribution of ?, the average number of events
per unit interval is
? Gamma(x , 1/t )
a distribution of the time until the next x
events have occurred
tx Gamma(x , 1/l ?)
When x 1, then exponential tnext Gamma(1 ,
1/l ?) Expon (1/?)
Thus the lower bound for b is also given by 1/
tnext 1/Gamma(1 , 1/l ?) 1/Expon (1/?)

75
Poisson Process-for unknown parameters

IF the assumptions of the Poisson process can be
satisfied

x Poisson (lt)
? Gamma(x , 1/t )
tx Gamma(x , 1/l ?)
Berhanu Tameru, CCEBRA, Tuskegee University
76
The Poisson Process

A Poisson process has the following
characteristics
The number of bacteria per liter of water (per
area of dish).
The number of outbreaks of disease per year.
The number of earth quakes per decade.

77
The Poisson Process

Historical information indicates an outbreak
occurs on average every 24 months. b 24, l
1/24 0.04 outbreaks per month. The number of
outbreaks in the next 6 months is then modelled
as Poisson (60.04)
If we observed 3 outbreaks over a period of 36
months we could estimate that the average number
of outbreaks per month is ? Gamma(x , 1/t )
Gamma ( 36, 1/3)
Probability of at least one event in an interval
P(x 1) 1 Exp ( -t/b).
What is the probability of at least one outbreak
during next 6 months given that the mean interval
between outbreaks is 24 months.
P(x 1) 1 Exp ( -6/24) 0.22