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
• Determine the basic concepts and basic laws of
  probability
• Find the probability of an event
• Solve the probability problems by axioms of
  probability
• Find the conditional probability of an event
• Find the probability of independents events
• Solve probability problems by using the Bayes
  Theorem

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

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Basic Concepts

• A probability experiment is a chance process that
  leads to well-defined results called outcomes.
• Ex Toss a coin
• An outcome is the result of a single trial of a
  probability experiment.
• Ex Head, Tail
• A sample space is the set of all possible
  outcomes of a probability experiment.
• Ex Head, Tail
• An event consists of a subset or collection of
  outcomes from the sample space.
• Ex Head, Tail, Head, Tail
• A simple event is an individual outcome from the
  sample space
• Head, P(Head)1/2

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Example 1

• Find the sample space for the following
  probability experiments
• a) Roll a die
• b) Answer a true/false question
• c) Toss 2 coins
• d) Gender of the children if a family has
• 3 children

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Tree Diagram

• A tree diagram is a device consisting of line
  segments emanating from a starting point and also
  the outcome point.
• It is used to determine all possible outcomes of
  a probability experiment

outcomes
AA
A
A
B
AB
A
BA
B
B
BB
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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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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.
• Equally likely events are events that have the
  same probability of occurring.
• A and B are equally likely event if P(A) P(B)
• For any experiment and any event of A, thus the
  probability that the event A occurs, P(A) is
  given by

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Example 2

• A single die is rolled. Find the probability of
  getting number 5.
• A coins is tossed. Find the probability of
  getting head.
• If a family has 3 children, find the probability
  that all the children are boys.

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Example 3

• If two dice are rolled one time, find the
  probability of getting these result
• A sum of 6
• Doubles
• A sum greater than 9

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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 4

• 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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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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Example 5

• 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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Basic Probability Rules

• Let S be a sample then P (S) 1
• For any event A, and
• For any event A, where
  is the set of outcomes in the sample
  space that are not included in the outcomes of
  event A.
• Venn diagrams are used to represent probabilities
  pictorially.

S
P(A)
P(A)
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Example 6

• 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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Basic Probability Rules

• 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
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Addition Rules for Probability

• The events A and B are said to be mutually
  exclusive event if they have no outcomes in
  common
• If A and B are mutually exclusive events, then
  
• If A and B are mutually exclusive events, then
• A collection of events are
  said to be mutually exclusive if no two of them
  have any outcome in common
• More generally, if are
  mutually exclusive, then

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Example 7

• A box contains 3 strawberry doughnuts, 4 jelly
  doughnuts and 5 chocolate doughnuts. If a person
  selects a doughnut at random, find the
  probability that it is either a strawberry
  doughnuts or chocolate doughnuts.
• A day of the week is selected at random. Find the
  probability that it is a weekend day.
• In a hospitals unit, there are 8 nurses and 5
  physicians. 7 nurses and 3 physicians are
  females. If a staff person is selected, find the
  probability that the staff is a nurse or a male.
• On New Years eve, the probability of a person
  driving while intoxicated is 0.32, the
  probability of a person having a driving accident
  is 0.09, and the probability of a person having a
  driving accident while intoxicated is 0.06. What
  is the probability of a person driving while
  intoxicated or having a driving accident?

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3.2 Conditional Probability

• The conditional probability of an event B in
  relationship to an event A was defined as the
  probability that event B occurs after event A has
  already occurred. That is P(BA).

Motivation 100 applicants for a post of lecturer
in UMP 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 8
Given
Find i.
ii.
iii.
iv.
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Example 9

• When a die 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 )
, P(A and B) P(BA)P(A)
B

P(A)
A
B
P(BA)
, P(A and B) P(BA)P(A)


P(BA)
, P(A and B) P(BA)P(A)
B
A
P(A)

P(BA)
,P(A and B) P(BA)P(A)
1
B
1
P(A) P(A and B) P(A and B)
P(B) P(B and A) P(B and A)
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Example 10
Event X and Y are such that
By drawing a tree diagram, find i.
ii.
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Example 11

• 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, find the probability that both are
  green?
• Use tree diagram

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3.4 Independent Event
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Example 12
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Example 13
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3.4 Bayes Theorem
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Example 14

• 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 15
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