PRED 354 TEACH. PROBILITY

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PRED 354 TEACH. PROBILITY STATIS. FOR PRIMARY
MATH

• Lesson 5
• PROBABILITY 

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CORRECTIONS

• 1. Central tendency Mean, Median Mode

nominal
There are a few extreme scores in the
distribution Some scores have undetermined
values There are open ended distribution The data
measured on an ordinal scale
There are a few extreme scores in the
distribution Some scores have undetermined
values There are open ended distribution The data
measured on an ordinal scale
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CORRECTIONS

• MISCONCEPTION
• Many of you wrote for the sample A, the
  semi-interquartile range is more appropriate,
  because the semi-interquartile of sample A is
  smaller than that of sample B.
• Two samples are as follows
• Sample A 7, 9, 10, 8, 9, 12
• Sample B 13, 5, 9, 1, 17, 9

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CORRECTIONS

• 2. Variability Range, Semi-interquartile range,
  variance, standard deviation

1.Extreme scores. 2. Sample size. 3.
Stability under sampling 4. Open-ended
distributions
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CORRECTIONS

• Calculating sample standard deviation

Population Sample
Mean µ X
variance s2 SS/N s2SS/n-1
Standard deviation s vSS/N s vSS/n-1
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Interpretations of Probability

• The frequency interpretation of probability
• The probability that some specific outcome of a
  process will be obtained can be interpreted to
  mean the relative frequency with which that
  outcome would be obtained if the process were
  repeated a large number of times under similar
  conditions.

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Interpretations of Probability

• 2. The classical interpretation of probability
• It is based on the concept of equally likely
  outcomes.

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Interpretations of Probability

• 3. The subjective interpretation of probability
• The probability that a person assigns to a
  possible outcome of some process represents
  her/his own judgment of the likelihood that the
  outcome will be obtained. This judgment will be
  based on each persons beliefs or information
  about the process.
• It is appropriate to speak of a certian persons
  subjective probability , rather than to speak of
  the true probability of that outcome.

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Experiments

• An experiment is the process of making
  observation.
• Ex
• a. A coin is tossed 10 times. The experimenter
  might want to determine the probability that at
  least four heads will be obtained.
• b. In an experiment in which a sample of 1000
  transistors is to be selected from a large
  shipment of similar items and each selected item
  is to be inspected, a person might want to
  determine the probability that not more than one
  of the selected transistors will be defective.

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Sample space

• A sample space is a set of points corresponding
  to all distinctly possible outcomes of an
  experiment.
• Ex For the die tossing experiment,

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Sample point

• A sample point is a point in a sample space.
• Ex For the die tossing experiment,

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Descrete sample

• A descrete sample space is one that contains a
  finite number or countable infinity of sample
  points.
• Ex A coin is tossed two times.

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Event

• For a descrete sample space, an event is any
  subset of it.
• Ex a. A coin is tossed two times.
• b. For the die tossing experiment
• Note simple event
• Ex observe a 6.

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Summarizing example

• Tossing a Coin Suppose that a coin is tossed
  three times. Then
• Experiment
• Sample space
• Sample point
• Events
• Simple event

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Definition of probability

• Axiom 1. For every event A, Pr (A)0.
• Axiom 2. Pr (S) 1.
• Axiom 3 For every infinite sequence of disjoint
  events

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Theorem 1
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Theorem 2

• For every finite sequence of n disjoint events

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Theorem 3

• For every event A

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Theorem 4

• If , then

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Theorem 5

• For every event A,

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Theorem 6

• For every two events A and B,

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Summarizing example

• Diagnosing Diseases A patient arrives at a
  doctors office with a sore throat and low grade
  fever. After an exam, the doctor decides that the
  patient has either a bacterial infection or a
  viral infection or both. The doctor decides that
  there is a probability of 0.7 that the patient
  has a bacterial infection and a probability of
  0.4 that the person has a viral infection. What
  is the probability that the patient has both
  infection?

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Summarizing example 2

• Demands for Utilities A contractor is building
  an office complex and needs to plan for water and
  electricity demands (sizes of pipes, conduit, and
  wires). After consulting with prospective tenants
  and examining historical data, the contractor
  decides that the demand for electricity will
  range between 1 million and 150 million
  kilowatt-hours per day and water demand will be
  between 4 and 200 (in thousand gallons per day).
  All combinations of ellectrical and water demand
  are considered possible.

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Finite sample space

• Experiments include a finite number of possible
  outcomes.
• The number is the probability that the
  outcome of the experiment will be

If the probability assigned to each of the
outcomes is 1/n, then this sample space S is a
simple sample space.
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Summarizing example

• Fiber breaks consider an experiment in which
  five fibers having different lenghts are
  subjected to a testing process to learn which
  fiber will break first. Suppose that the lenghts
  of the five fibers are 1, 2, 3, 4, and 5 meters,
  respectively. Suppose also that probability that
  any given fiber will be the first to break is
  proportional to the lenght of that fiber.
  Determine the probability that the lenght of the
  fiber that breaks first is not more than 3
  meters.

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The probability of a union of events

• If the events are disjoint,
• Theorem For every three events,

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Summarizing example

• Student Enrollment Among a group of 200
  students, 137 students are enrolled in a
  mathemtical class, 50 students are enrolled in a
  history class, and 124 students are enrolled in a
  music class. Furthermore, the number of students
  enrolled in both the mathematics and history
  classes is 33 the number enrolled in both the
  history and music class 29, and the number
  enrolled in both the methemtics and music class
  is 92. Finally, the number of students enrolled
  in all three classes is 18. Determine the
  probability that a student slected at random from
  the group of 200 stundents will be enrolled in at
  least one of the three classes.

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Teaching probability

• Constructing probability examples
• Work with examples such as the probability of boy
  and girl births and use probability models of
  real outcomes.
• These are more interesting and are known than
  card and crap games.

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Teaching probability

• Random numbers via dice or handouts
• Rolling the dice ones gives a random digit.
• If it is too inconvenient, you can prepare
  handouts of random numbers for your students.
• You can use already existing material.
• Ex telephone book.

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Teaching probability

• Probability of compound events
• Use babies or real vs. fake coin flips.
• Babies Students enjoy examples involving
  families and babies.
• EX We adapt a standard problem in probability by
  asking students which of the following sequences
  of boy and girl births is most likely, given that
  a family has four children bbbb, bgbg, or gggg.

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Teaching probability

• Probability of compound events
• Real vs. fake coin flips Students often have
  diffuculties with probability of distributions.
• We pick two students to be judges and one to be
  the recorder and divide the others in the class
  into two groups.
• One group is instructed to flip a coin 100
  times, or flip 10 coins 10 times each, or follow
  some similarly defined protocol, and then to
  write the results, in order, on a sheet of paper,
  writing heads as 1 and tails as 0.
• The second group is instructed to create a
  sequence of 100 0s ans 1s that are intended
  to look like the result of coin flips- but they
  are to do this without flipping any coins or
  randomization device- and to write this sequence
  on a sheet of paper.

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Teaching probability

• Probability modeling
• We can apply probabilty distributions to real
  phenomena.
• Ex Airplane failure (and other rare events)
• Looking back historical data gave probability
  estimate of about 2. Its deadly accident was
  calculated as 82. what is the probabilty of that
  a person will be dead in an airplane accident due
  to airplane failure?