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MAT271E - Probability and Statistics
Counting Techniques, Concept of Probability, Probability Function, Probability
Density Function, Bernoulli, Binom, Poisson Disributions, Exponantial, Gamma,
Normal Density Functions, Random Variables of Multiple Dimensions, The
Concept of Estimator and Properties of Estimators, Maxsimum Likelihood
Function, Test of Hypothesis, Ki-Square Test, t-test, F-test, Correlation Theory.
Exam
a)Midterm Ex
b)Quiz
c)Homework
d)Final
Grade
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Terminology:
– Trial: each time you repeat an experiment
– Outcome: result of an experiment
– Random experiment: one with random
outcomes (cannot be predicted exactly)
– Relative frequency: how many times a specific
outcome occurs within the entire experiment.
– Sample space: the set of all possible outcomes of
an experiment
– Event: any subset of the sample space
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In a number of different situations, it is not easy
to determine the outcomes of an event by
counting them individually.
Alternatively, counting techniques that involve
permutations
And
combinations
are helpful when calculating theoretical
probabilities.
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Counting principle states that "If there
are m ways to do one thing, and n ways to do
another, and t ways to do a third thing, and so on
..., then the number of ways of doing all those
things at once is m x n x t x ...
Let's look at an actual example and try to make
sense of this rule. How about a license plate ...
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How many different license plates are there altogether?
Look at what's used to make a plate:
LETTER LETTER LETTER NUMBER NUMBER NUMBER
For each of the letters we have 26 choices. For each of
the numbers we have 10 choices.
The Fundamental Counting Principle says that:
The total number of ways to fill the six spaces on a licence
plate is
26 x 26 x 26 x 10 x 10 x 10
which equals
17,576,000
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GZPA74
How could the province increase the total
number of possible license plates? One way
would be to make the plates with four letters
and two numbers. Then the total number of
plates would be:
26 x 26 x 26 x 26 x 10 x 10 = 45,697,600
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Tree Diagrams
Tree diagrams are a graphical way of listing all
the possible outcomes.
The outcomes are listed in an orderly fashion, so
listing all of the possible outcomes is easier than
just trying to make sure that you have them all
listed
• When calculating probabilities, you need
to know the total number of outcomes in
the sample space
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Use a TREE DIAGRAM to list the sample
space of 2 coin flips.
Sample
Space
H
If you got H
Now you could get…
H
T
YOU
On the first flip you
could get…..
H
T
If you got T
Now you could get…
T
T
The final outcomes are obtained by following each branch to its conclusion: They are
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from top to bottom: HH HT TH TT
Multiplication Rule of Counting
• The size of the sample space is the
denominator of our probability
• So we don’t always need to know what
each outcome is, just the number of
outcomes.
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Multiplication Rule of
Compound Events
If…
• X = total number of outcomes for event A
• Y = total number of outcomes for event B
• Then number of outcomes for A followed by
B = x times y
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Multiplication Rule:
Dress Mr. Arnold
• Mr. Reed had 3 EVENTS
2
shoes
2
pants
3
shirts
How many outcomes are there
for EACH EVENT?
2(2)(3) = 12 OUTFITS
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• Sometimes we are concerned with
how many ways a group of objects
can be arranged
•How many ways to arrange books on a shelf
•How many ways a group of people can stand in line
•How many ways to scramble a word’s letters
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If we have n distinct objects and we
want to put them in some sort of
ordered set (arrangement) we use
permutations, denoted n Pn or P  n, n 
and equal to
n
!

n
n

1
n

2
n

3
.
.
.
2

1






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5!
• denoted with ! 
• Multiply all integers ≤ the number 
5! = 5(4)(3)(2)(1) = 120
• 0! =
1
• 1! =
1
• Calculate 6!
6! = 6(5)(4)(3)(2)(1) = 720
• What is 6! / 5!?
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• denoted with !  5!
• Multiply all integers ≤ the number 
1
• 0! =
1
• 1! =
• Calculate 6!
• What is 6! / 5!?
6(5)(4)(3)(2)(1)
5(4)(3)(2)(1)
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=6
Harder things to count
Suppose that we still have n objects in our set
but that some of them are indistinguishable
from one another (an example of this would
be the set of letters in the word “googol”).
Suppose that we are interested in using all of
the letters for each arrangement but we want
to know how many distinguishably different
arrangements there can be.
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Here we want to use permutations again but
we want to divide out those permutations
which are not distinguishable from one
another by virtue of their containing some
of the repeated objects in different
locations. Suppose there are p of one
object, q of another, etc. Our calculation can
be done as
n!
p !q !
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• You have n objects
• You select r objects
• This is the number of ways you
could select and arrange in
order:
n!
n
Pr =
(n− r )!
Another common notation for a permutation is nPr
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• Sometimes, we are only concerned with
selecting a group and not the order in
which they are selected.
• A combination gives the number of
ways to select a sample of r objects
from a group of size n.
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• You have n objects
• You want a group of r object
• You DON’T CARE what order they are
selected in
n
Cr =
n!
r!(n− r )!
Combinations are also denoted nCr
Read “n choose r”
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• Order matters Permutation
• Order doesn’t matter  Combination
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• Many things in everyday life, from stock price to
lottery, are random phenomena for which the
outcome is uncertain.
• The concept of probability provides us with the
idea on how to measure the chances of possible
outcomes.
• Probability enables us to quantify uncertainty,
which is described in terms of mathematics
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• what is the chance that a given event will occur?
For us, what is the chance that a child, or a
family of children, will have a given phenotype?
• Probability is expressed in numbers between 0
and 1. Probability = 0 means the event never
happens; probability = 1 means it always
happens.
• The total probability of all possible event always
sums to 1.
0⪯P⪯1
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• The probability of an event equals the number of
times it happens divided by the number of
opportunities.
• These numbers can be determined by
experiment or by knowledge of the system.
The Number Of Ways Event A Can Occur
Probability Of An Event P(A) = ---------------------The total number Of Possible Outcomes
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Four people run in a marathon. In how many ways
can first and second place finish?
Hasan, Ayşe, Ali, Veli
Ayşe
Hasan
Ali
Veli
Ayşe
Hasan
Ali
Veli
Hasan
Ali
Ayşe
Veli
Hasan
Veli
Ayşe
Ali
4.3=12 Outcomes
Event 2 occurs 3 ways
Event 1 occurs 4 ways
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Manipulating Numbers
“There are lies, damned lies, and statistics”
-- Disraeli
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Anecdotal evidence is unreliable
Why does the phone always ring when you’re in
the shower?
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Determining the difference between chance
and real effects
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Data = Signal + Noise
Signal = What we’re trying to measure
Noise = Error in our measurement
If noise is random, then as the sample
size increases, noise tends to cancel,
leaving only signal.
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• Statistics is the area of science that deals with
collection, organization, analysis, and interpretation
of data.
• It also deals with methods and techniques that can
be used to draw conclusions about the
characteristics of a large number of data points-commonly called a population-• By using a smaller subset of the entire data.
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For Example…
• You work in a cell phone factory and are asked to
remove cell phones at random off of the assembly
line and turn it on and off.
• Each time you remove a cell phone and turn it on
and off, you are conducting a random experiment.
• Each time you pick up a phone is a trial and the result
is called an outcome.
• If you check 200 phones, and you find 5 bad phones,
then
• relative frequency of failure = 5/200 = 0.025
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Statistics in Engineering
• Engineers apply physical
and chemical laws and
mathematics to design,
develop, test, and supervise
various products and
services.
• Engineers perform tests to
learn how things behave
under stress, and at what
point they might fail.
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Statistics in Engineering
• As engineers perform experiments, they
collect data that can be used to explain
relationships better and to reveal information
about the quality of products and services
they provide.
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