Transcript Lecture 1 - Statistics

Today
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Today: Course Outline, Start Chapter 1
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Assignment 1:
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Read Chapter 1 by next Tuesday
Statistics
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What is Statistics?
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Discipline that deals with the collection, summary, organization and
interpretation of data
Used to help predict and answer questions about real processes
Example
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Dr. B. Spock was convicted of conspiracy during Vietnam era
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Convicted by an all-male jury
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Felt trial was unfair because no women were on the jury
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Jury list for Spock’s judge had 14.4% women
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Jury list for 6 other judges had 29% women
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Does this seem fair?
Example
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Questions:
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What is the chance or probability that in a population with 29% women,
we could select a jury pool of only 14.4% women?
Is 14.4% close to 29%?
What does chance or probability mean?
Sample Space
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The term experiment is used to denote an activity which produces an
unpredictable outcome
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Will describe and summarize experiments using models
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Model describing an experiment should to take into account all
possible outcomes
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Collection of all possible outcomes of an experiment is called the
sample space and is denoted by 
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Example
A balanced coin is flipped
Do not know outcome in advance
 {
}
What if we tossed the coin 2 times?
 {
}
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Example
Roll of a fair die
 {
}
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Example
Number of people arriving in an emergency room in before one dies
 {
}
Event
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A collection of outcomes (a subset of the sample space) is called an
event
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An event, E, is said to occur if one of its outcomes occurs
Event
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Example
A balanced coin is flipped two times
 {
}
Let E be the event that a tails occurs
E={
}
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Example
Committee of two people is chosen from 5 people (a-e)
 {
}
Let E be the event that person a is on the committee
E={
}
Event Algebra
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We will generally introduce these ideas as needed
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A  B  outcomes in A or B (Union)
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AB  outcomes in A and B (Intersection)
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Ac  Everything in the sample space that is not in A
Event Algebra
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Example: Fair Die
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  {1,2,3,4,5,6}
E1  {Even outcome}  {2,4,6}
E2  {Odd outcomes}  {1,3,5}
E3  {1,2,3}
E1E3={
}
E2  E3  {
E1E2  {
E3c  {
}
}
}
Venn Diagrams
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Can represent sample space and events using a Venn Diagram
Probability for Experiments
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A probability model assigns probabilities to outcomes and/or events
in the sample space
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Simplest case is when the number of possible outcomes in the
sample space is finite and each outcome is equally likely
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If there are N outcomes in the sample space and each outcome is
equally likely then the probability of any individual outcome is 1/N
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Example: Fair die is rolled. What is the probability of observing 6?
Probability for Experiments
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An event E is a collection of outcomes from the sample space
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Recall, an event, E, is said to occur if one of its outcomes occurs
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So, P(E)=
Probability for Experiments
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Example:
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In roulette, the possible outcomes are {00,0,1,2,…36}
Outcomes 00 and 0 are green, odd outcomes are red and even outcomes
are black
Let R be the event that a red is observed in one spin of the roullette
wheel
P(R)=
Example:
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In the committees example, the probability that person a is on the
committee is…
Properties of Probability
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0  P( E )  1; for any event E
P()  1
P( E )  P( F )  P( E )  P( F ) whenever E and F are mutually exclusive
These are called the axioms of probability
Consequences of the Axioms
c
• P( E )  1  P( E )
• P( E  F )  P( E )  P( F )  P( E  F )
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Example:
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Small university offers 2 language courses, French and German
Everyone is required to take a language course
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If 75% of students take French and 55% take German what proportion
of students take both courses?
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What proportion take only one course?
Combinatorics
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In the equally likely case, computing probabilities involves counting
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This can be hard…really
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Combinatorics is a branch of mathematics whichs develops efficient
counting methods
Combinatorics
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Consider the rhyme
As I was going to St. Ives
I met a man with seven wives
Every wife had seven sacks
Every sack had seven cats
Every cat had seven kits
Kits, cats, sacks and wives
How many were going to St. Ives?
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Answer:
Combinatorics
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