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MATH/STAT 425

Welcome!

TODAY: Introduction to the course

Start Chapter 1 material

For next time:

Read Ch 1, 2.1, 2.2

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Why probability?

Random process:

Examples:

Humans have poorly formed intuitions aboutrandom processes.

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Examples

Gambling

Red light/green light experiments (rat vs.

human)

Goal: guess the next light as many times as

possible

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More examples Birthday problem: What is the chance that at least two people

in a room share a birthday (the daynot necessarily the year)

if there are 20 people in the room? 40 people? 60 people? 70

people?

In this room, there are about 50 people.

# people 20 40 60 70

probability

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More examplesAIDS testing: Suppose 99% of people with AIDS test positive,

95% of people without AIDS test negative, and .1% of people

actually have AIDS. What is the chance that a person who

tests positive actually has AIDS?

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This course

GOAL: Introduce a formal mathematicalframework to better understand randomprocesses.

STRUCTURE:

Lecture 2 times/week

Roughly one assignment per week

2 midterm exams

Final exam

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Chapter 1Combinatorial Analysis

Combinatorial Analysis:

With combinatorial analysis, we can

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Outline of Chapter 1

Basic principle of counting (BPC)

Permutations

Combinations

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Basic Principle of Counting

Ex1.1: A small community consists of 10 women,EACH of whom has 3 children. If one woman and

one of her children are to be chosen as mother and

child of the year, how many different choices are

possible? (Hint: Draw a diagram)

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Theorem: Two experiments

Suppose twoexperiments will be performed.

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Example: More than two experiments

Ex1.2: A college planning committee consists of 3

freshman, 4 sophomores, 5 juniors, and 2 seniors. A

subcommittee of 4 consisting of 1 person from each

class is to be chosen. How many differentsubcommittees are possible?

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A General Theorem (BPC)

Supposer experiments will be performed.

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Examples

Ex1.3: Suppose license plates have numbers in the first3 places followed by 3 letters.

(a) If letters and numbers and can be repeated, how

many different license plates are possible?

(b) What about without repetition?

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Permutations

Ex1.4:How many different ordered arrangements ofthe letters a, b, care possible?

Permutation:

In general, suppose we have nobjects, then thereare

different permutations of the nobjects.

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More Examples

Ex1.5: How many different batting orders are possible for a

team consisting of 9 players?

Ex1.6: A class consists of 6 men and 4 women. Anexamination is given, and the students are rankedaccording to their performance.

(a) How many different rankings are possible?

(b) If the men are ranked just among themselves, and thewomen are ranked just among themselves, how manydifferent rankings are possible?

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Undistinguishable permutations

Question: How many distinct arrangementscan be formed from the letters

P O P?

Answer:

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Undistinguishable permutations

Question: How many distinct arrangements

can be formed from the letters

P E P P E R ?

Answer:

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Another example

Ex1.7: A chess tournament has 10 competitors, ofwhich 4 are Russian, 3 are from the US, 2 are from

Great Britain, and 1 is from Brazil. If the tournament

result lists just the nationalities of the players in the

order in which they place, how many differentoutcomes are possible?

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Combinations

Ex1.8: A committee of 3 is to be formed from a

group of 5 people. How many different

committees are possible?

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Combinations

Combination:

Notation:

In general,

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Examples

Ex1.9: Senatorial committees:(a) How many committees with 2 senators can be formed from

a group of 5 senators?

(b) How many committees with 2 Republican, 2 Democrat, and

3 Independent senators can be formed from a group of 5

Republican, 6 Democrat, and 4 Independent senators?

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Another examplethink about it for

next time!

Ex1.10: A communication system is to consist of nantennasthat are to be lined up in a linear order. The resulting system

is said to be able to receive all incoming signalsand be

called functionalas long as no two consecutive antenna are

defective. If it turns out that exactly mof the nantennas aredefective, how many possible configurations of the antennas

will make the system work?

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Next time:

Finish Chapter 1

First HW assigned Tuesday (1/9) and due

Tuesday 1/16

READ: Chapter 1.