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Basic Probability
IBS-09-SL RM 501 – Ranjit Goswami
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Introduction
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Probability is the study of randomness and uncertainty.
In the early days, probability was associated with games
of chance (gambling).
IBS-09-SL RM 501 – Ranjit Goswami
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Simple Games Involving Probability
Game: A fair die is rolled. If the result is 2, 3, or 4, you win
$1; if it is 5, you win $2; but if it is 1 or 6, you lose $3.
Should you play this game?
IBS-09-SL RM 501 – Ranjit Goswami
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Random Experiment
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a random experiment is a process whose outcome is uncertain.
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Examples:
Tossing a coin once or several times
Picking a card or cards from a deck
Measuring temperature of patients
...
IBS-09-SL RM 501 – Ranjit Goswami
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Events & Sample Spaces
Sample Space
The sample space is the set of all possible outcomes.
Simple Events
The individual outcomes are called simple events.
Event
An event is any collection
of one or more simple events
IBS-09-SL RM 501 – Ranjit Goswami
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Example
Experiment: Toss a coin 3 times.
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Sample space 
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 = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
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Examples of events include
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A = {HHH, HHT,HTH, THH}
= {at least two heads}
B = {HTT, THT,TTH}
= {exactly two tails.}
IBS-09-SL RM 501 – Ranjit Goswami
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Basic Concepts (from Set Theory)
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The union of two events A and B, A  B, is the event consisting of
all outcomes that are either in A or in B or in both events.
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The complement of an event A, Ac, is the set of all outcomes in 
that are not in A.
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The intersection of two events A and B, A  B, is the event
consisting of all outcomes that are in both events.
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When two events A and B have no outcomes in common, they are
said to be mutually exclusive, or disjoint, events.
IBS-09-SL RM 501 – Ranjit Goswami
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Example
Experiment: toss a coin 10 times and the number of heads is observed.
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Let A = { 0, 2, 4, 6, 8, 10}.
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B = { 1, 3, 5, 7, 9}, C = {0, 1, 2, 3, 4, 5}.
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A  B= {0, 1, …, 10} = .
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A  B contains no outcomes. So A and B are mutually exclusive.
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Cc = {6, 7, 8, 9, 10}, A  C = {0, 2, 4}.
IBS-09-SL RM 501 – Ranjit Goswami
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Rules
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Commutative Laws:
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A  B = B  A, A  B = B  A
Associative Laws:
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(A  B)  C = A  (B  C )
(A  B)  C = A  (B  C) .
Distributive Laws:
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(A  B)  C = (A  C)  (B  C)
(A  B)  C = (A  C)  (B  C)
IBS-09-SL RM 501 – Ranjit Goswami
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Venn Diagram

A
A∩B
B
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Probability
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A Probability is a number assigned to each subset (events) of a sample
space .
Probability distributions satisfy the following rules:
IBS-09-SL RM 501 – Ranjit Goswami
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Axioms of Probability
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For any event A, 0  P(A)  1.
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P() =1.
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If A1, A2, … An is a partition of A, then
P(A) = P(A1) + P(A2) + ...+ P(An)
(A1, A2, … An is called a partition of A if A1  A2  … An = A and A1,
A2, … An are mutually exclusive.)
IBS-09-SL RM 501 – Ranjit Goswami
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Properties of Probability
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For any event A, P(Ac) = 1 - P(A).
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If A  B, then P(A)  P(B).
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For any two events A and B,
P(A  B) = P(A) + P(B) - P(A  B).
For three events, A, B, and C,
P(ABC) = P(A) + P(B) + P(C) P(AB) - P(AC) - P(BC) + P(AB C).
IBS-09-SL RM 501 – Ranjit Goswami
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Example
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In a certain population, 10% of the people are rich, 5% are famous,
and 3% are both rich and famous. A person is randomly selected
from this population. What is the chance that the person is
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not rich?
rich but not famous?
either rich or famous?
IBS-09-SL RM 501 – Ranjit Goswami
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Intuitive Development
(agrees with axioms)
• Intuitively, the probability of an event a could be
defined as:
Where N(a) is the number that event a happens in n trials
IBS-09-SL RM 501 – Ranjit Goswami
Here We Go Again: Not So Basic
Probability
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More Formal:
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 is the Sample Space:
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Contains all possible outcomes of an experiment
w in  is a single outcome
A in  is a set of outcomes of interest
IBS-09-SL RM 501 – Ranjit Goswami
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Independence
• The probability of independent events A, B and C is
given by:
P(A,B,C) = P(A)P(B)P(C)
A and B are independent, if knowing that A has happened
does not say anything about B happening
IBS-09-SL RM 501 – Ranjit Goswami
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Bayes Theorem
• Provides a way to convert a-priori probabilities to aposteriori probabilities:
IBS-09-SL RM 501 – Ranjit Goswami
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Conditional Probability
• One of the most useful concepts!

A
B
IBS-09-SL RM 501 – Ranjit Goswami
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Bayes Theorem
• Provides a way to convert a-priori probabilities to aposteriori probabilities:
IBS-09-SL RM 501 – Ranjit Goswami
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Using Partitions:
• If events Ai are mutually exclusive and partition 
IBS-09-SL RM 501 – Ranjit Goswami
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Random Variables
• A (scalar) random variable X is a function that maps
the outcome of a random event into real scalar
values

X(w)
w
IBS-09-SL RM 501 – Ranjit Goswami
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Random Variables Distributions
• Cumulative Probability Distribution (CDF):
• Probability Density Function (PDF):
IBS-09-SL RM 501 – Ranjit Goswami
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Random Distributions:
• From the two previous equations:
IBS-09-SL RM 501 – Ranjit Goswami
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Uniform Distribution
• A R.V. X that is uniformly distributed between x1 and
x2 has density function:
X1
X2
IBS-09-SL RM 501 – Ranjit Goswami
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Gaussian (Normal) Distribution
• A R.V. X that is normally distributed has density
function:
m
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Statistical Characterizations
• Expectation (Mean Value, First Moment):
•Second Moment:
IBS-09-SL RM 501 – Ranjit Goswami
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Statistical Characterizations
• Variance of X:
• Standard Deviation of X:
IBS-09-SL RM 501 – Ranjit Goswami
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Mean Estimation from Samples
• Given a set of N samples from a distribution, we can
estimate the mean of the distribution by:
IBS-09-SL RM 501 – Ranjit Goswami
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Variance Estimation from Samples
• Given a set of N samples from a distribution, we can
estimate the variance of the distribution by:
IBS-09-SL RM 501 – Ranjit Goswami
Pattern
Classification
Chapter 1: Introduction to Pattern
Recognition (Sections 1.1-1.6)
• Machine Perception
• An Example
• Pattern Recognition Systems
• The Design Cycle
• Learning and Adaptation
• Conclusion
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Machine Perception
• Build a machine that can recognize patterns:
• Speech recognition
• Fingerprint identification
• OCR (Optical Character Recognition)
• DNA sequence identification
IBS-09-SL RM 501 – Ranjit Goswami
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An Example
• “Sorting incoming Fish on a conveyor according to
species using optical sensing”
Sea bass
Species
Salmon
IBS-09-SL RM 501 – Ranjit Goswami
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• Problem Analysis
• Set up a camera and take some sample images to extract
features
• Length
• Lightness
• Width
• Number and shape of fins
• Position of the mouth, etc…
• This is the set of all suggested features to explore for use in our
classifier!
IBS-09-SL RM 501 – Ranjit Goswami
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Preprocessing
• Use a segmentation operation to isolate fishes from one
another and from the background
• Information from a single fish is sent to a feature
extractor whose purpose is to reduce the data by
measuring certain features
• The features are passed to a classifier
IBS-09-SL RM 501 – Ranjit Goswami
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• Classification
• Select the length of the fish as a possible feature for
discrimination
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The length is a poor feature alone!
Select the lightness as a possible feature.
IBS-09-SL RM 501 – Ranjit Goswami
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IBS-09-SL RM 501 – Ranjit Goswami
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• Threshold decision boundary and cost relationship
• Move our decision boundary toward smaller values of
lightness in order to minimize the cost (reduce the number
of sea bass that are classified salmon!)
Task of decision theory
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• Adopt the lightness and add the width of the fish
Fish
xT = [x1, x2]
Lightness
Width
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• We might add other features that are not correlated
with the ones we already have. A precaution should be
taken not to reduce the performance by adding such
“noisy features”
• Ideally, the best decision boundary should be the one
which provides an optimal performance such as in the
following figure:
IBS-09-SL RM 501 – Ranjit Goswami
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• However, our satisfaction is premature because
the central aim of designing a classifier is to
correctly classify novel input
Issue of generalization!
IBS-09-SL RM 501 – Ranjit Goswami
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IBS-09-SL RM 501 – Ranjit Goswami