DCT2063 CHAPTER THREE PROBABILITY

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DCT2063CHAPTER THREEPROBABILITY
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Content

• 3.1 Basic Idea and Consideration
• 3.2 Conditional Probability
• 3.3 Independent Event
• 3.4 Bayes Theorem

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OBJECTIVES

• After completing this chapter, you should be able
  to
• Understand the basic concepts and basic laws of
  probability
• Find the probability of an event
• Solve the probability problems
• Understand conditional and independent events
• Find the probability of conditional and
  independents events
• Understand and use the Bayes Theorem to solve
  probability problems

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3.1 Basic Idea Consideration

• Probability as a general concept can be defined
  as the chance of an event occurring. In addition
  to being used in games of chance, probability is
  used in the fields of insurance, investments, and
  weather forecasting, and in various areas.
• Rules such as the fundamental counting rule,
  combination rule and permutation rules allow us
  to count the number of ways in which events can
  occur.
• Counting rules and probability rules can be used
  together to solve a wide variety of problems.

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Basic Concepts

• A probability experiment is a chance process that
  leads to well-defined results called outcomes.
• An outcome is the result of a single trial of a
  probability experiment.
• A sample space is the set of all possible
  outcomes of a probability experiment.
• An event consists of a subset or collection of
  outcomes from the sample space.
• A simple event is an individual outcome from the
  sample space

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Basic Concepts (contd.)

• Venn diagrams are used to represent probabilities
  pictorially.
• Equally likely events are events that have the
  same probability of occurring.

S
P(A)
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Basic Concepts (contd.)

• Combining Events

Union - is the set of outcomes that
belong either to A or B
Intersection - is the set of
outcomes that belong to
both A and B
Complement - is the set of outcomes that do
not belong to A
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Basic Concepts (contd.)

• Mutually Exclusive Events

• The events A and B are said to be
• mutually exclusive event if they have
• no outcomes in common
• A collection of events
• is said to be mutually exclusive
• if no two of them have
• any outcome in common.

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Axioms of Probability

• Let S be a sample then

• For any event A,

• If A and B are mutually exclusive events, then
  
• More generally, if
  are mutually exclusive, then

• For any event A,

• If S is a sample space containing N equally
  likely outcomes,
• and if A is an event containing k outcomes,
  then

• Let A and B be any events, then

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3 Basic types of Probability

• Classical probability
• Empirical or frequency probability
• Subjective probability

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Classical Probability

• Classical probability uses sample spaces to
  determine the numerical probability that an event
  will happen.
• Classical probability assumes that all outcomes
  in the sample space are equally likely to occur
• For any experiment and any event of A, thus the
  probability that the event A occurs,
  is given by

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Example

• If a family has 3 children, find the probability
  that all the children are boys.
• When a single die is rolled, find the probability
  of getting a 9.
• When a single die is rolled, what is the
  probability of getting a number less than 7?
• If the probability that a person lives in an
  industrialized country of the world is 1/5, find
  the probability that a person does not live in an
  industrial company.

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Empirical Probability

• Empirical probability relies on actual experience
  to determine the likelihood of outcomes.
• Given a frequency distribution, the probability
  of an event being in a given class is

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Example

• Hospital records indicated that maternity
  patients stayed in the hospital for the number of
  days shown in the distribution
• Number of days stayed Frequency
• 3
  15
• 4
  32
• 5
  56
• 6
  19
• 7
  5
• 
  127
• Find these probabilities.
• A patient stayed exactly 5 days
• A patients stayed less than 6 days
• A patient stayed at most 4 days
• A patient stayed at least 5 days

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Solves problems involving linear inequalities

• At least, minimum of, no less than
• At most, maximum of, no more than
• Is greater than, more than
• Is less than, smaller than

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Subjective Probability

• Subjective probability uses a probability value
  based on an educated guess or estimate, employing
  opinions and inexact information.
• In subjective probability, a person or group
  makes an educated guess at the chance that an
  event will occur. This guess is based on the
  persons experience and evaluation of a solution.

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Examples

• A seismology might say there is an 80
  probability that an earthquake will occur in a
  certain area
• A doctor might say that on the basis of his
  diagnosis, there is a 30 chance the patient will
  survive in an operation.

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3.2 Conditional Probability
Motivation 100 applicants for a post of lecturer
in KUKTEM are categorized through their gender
and experience.
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Formula for Conditional Probability

• The probability that the second event B occurs
  given that the first event A has occurred can be
  found dividing the probability that both events
  occurred by the probability that the first event
  has occurred. The formula is

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Example 1
Given
Find i.
ii.
iii.
iv.
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Examples

• When a dice was thrown, the score was an odd
  number. What is the probability that it was a
  prime number?
• A box contains black chips and white chips. A
  person selects 2 chips without replacement. If
  the probability of selecting a black chip and a
  white chip is 15/56 and the probability of
  selecting a black chip on the first draw is 3/8,
  find the probability of selecting the white chip
  on the second draw, given that the first chip
  selected was a black chip

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Tree Diagram
P(BA ) , AB
B
P(A)
A
B
P(BA) , AB
P(BA ) , AB
B
A
P(A)
P(BA ) , AB
B
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Example

• We have 10 pieces of candy in a dish. We know
  that 5 pieces is red, 3 are green, and 2 are
  yellow. If we choose 2 pieces at random without
  looking, whats the probability that both are
  green?

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Example
Event X and Y are such that
By drawing a tree diagram, find i.
ii.
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3.4 Independent Event
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Examples
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Examples
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3.4 Bayes Theorem
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Example 1

• The proportion of people in a given community who
  have a certain disease is 0.005. A test is
  available to diagnose the disease. If a person
  has the disease, the probability that the test
  will produce a positive signal is 0.99. If a
  person does not have the disease, the probability
  that the test will produce a positive signal is
  0.01 If a person tests positive, what is the
  probability that the person actually has the
  disease?

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Example 2
A record for a failed emission test is chosen at
random. Given, A1 Small engine car
A2 Medium engine car
A3 Large engine car
B failed emission test within 2 years
What is the probability that it is failed for a
car with a small engine?
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Conclusion

• Probability is the basis of inferential
  statistics
• Predictions are based on probability
• Hypothesis are tested by using probability

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Thank You