PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 5
PROBABILITY
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
CORRECTIONS
MISCONCEPTION:
Many of you wrote: “for the sample A, the semiinterquartile 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
CORRECTIONS
2. Variability: Range, Semi-interquartile range,
variance, standard deviation
1.Extreme scores.
2. Sample size.
3. Stability under sampling
4. Open-ended distributions
CORRECTIONS
Calculating sample standard deviation:
Population
Sample
Mean
µ
X
variance
σ2 = SS/N
s2=SS/n-1
Standard deviation
σ = √SS/N
s = √SS/n-1
Interpretations of Probability
1.
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.
Interpretations of Probability
2. The classical interpretation of probability
It is based on the concept of equally likely
outcomes.
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 person’s
beliefs or information about the process.
It is appropriate to speak of a certian person’s
subjective probability , rather than to speak of
the true probability of that outcome.
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.
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,
Sample point
A sample point is a point in a sample space.
Ex: For the die tossing experiment,
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.
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.
Summarizing example
Tossing a Coin: Suppose that a coin is tossed
three times. Then
Experiment :
Sample space :
Sample point:
Events:
Simple event:
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 Ai , A2 ,.....
   
Pr   Ai    Pr( Ai )
 i 1  i 1
Theorem 1
Pr     0
Theorem 2

For every finite sequence of n disjoint
events Ai , A2 ,.....
 n  n
Pr   Ai    Pr( Ai )
 i 1  i 1
Theorem 3

For every event A
Pr( A)  1  Pr( A)
Theorem 4

If A  B , then
Pr( A)  Pr( B)
Theorem 5

For every event A,
0  Pr( A)  1
Theorem 6

For every two events A and B,
Pr( A  B)  Pr( A)  Pr( B)  Pr( A  B)
Summarizing example
Diagnosing Diseases: A patient arrives at a
doctor’s 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?
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.
Finite sample space
Experiments include a finite number
possible outcomes.
of
S  s1 , s2 ,......., sn 
The number pi is the probability that the
outcome of the experiment will be si , (i  1, 2,3,...., n)
pi  0
n
p
i 1
i
1
If the probability assigned to each of the
outcomes is 1/n, then this sample space S is a
simple sample space.
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.
The probability of a union of events
If the events are disjoint,
 n  n
Pr   Ai    Pr( Ai )
 i 1  i 1
Theorem: For every three events,
Pr( A1  A 2  A3 )  ...
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.
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.
Teaching probability
Random numbers via dice or handouts
a.
Rolling the dice ones gives a random digit.
If it is too inconvenient, you can prepare handouts
of random numbers for your students.
c.
You can use already existing material.
Ex: telephone book.
b.
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.
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 “0”s ans “1”s 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.
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?