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MAT271E - Probability and StatisticsCounting 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.

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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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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:

GZPA74

26 x 26 x 26 x 26 x 10 x 10 = 45,697,600

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Tree Diagrams

• When calculating probabilities, you need to know the total number of outcomes in the sample space

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

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Use a TREE DIAGRAM to list the sample space of 2 coin flips.

YOU

On the first flip you could get…..

H

T

If you got HNow you could get…

If you got TNow you could get…

H

T

H

T

TSampleSpace

T

The final outcomes are obtained by following each branch to its conclusion: They are from top to bottom: HH HT TH TT

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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

pantsshoes shirts

How many outcomes are there for EACH EVENT?

2 2 3

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 shelfHow many ways to arrange books on a shelf

•How many ways a group of people can stand in lineHow many ways a group of people can stand in line

•How many ways to scramble a word’s lettersHow 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 or and equal to

n nP ,n nP

! 1 2 3 ...2 1n n n n n

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• denoted with ! • Multiply all integers ≤ the number

• 0! = • 1! = • Calculate 6!

• What is 6! / 5!?

5!

5! = 5(4)(3)(2)(1) = 12011

6! = 6(5)(4)(3)(2)(1) = 720

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• denoted with ! • Multiply all integers ≤ the number

• 0! = • 1! = • Calculate 6!

• What is 6! / 5!?

5!

5(4)(3)(2)(1)

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6(5)(4)(3)(2)(1) =6

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Harder things to countSuppose 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

• You select

• This is the number of ways you could select and arrange in order:

Prn=

n!(n−r )!

Another common notation for a permutation is nPr

n objectsr objects

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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

C rn=

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

Hasan AyşeAliVeli

Ayşe HasanAli

VeliAli Hasan

AyşeVeli

Veli HasanAyşe

Ali

4.3=12 Outcomes

Event 1 occurs 4 waysEvent 2 occurs 3 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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