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Probability

- The tool that allows the statisticians to use

sample information to make inferences about or to

describe the population from which the sample was

drawn

Some concepts

- Experiment is the process by which an

observation is obtained. - Examples
- - toss a die is a experiment
- - Draw a card from a deck of card
- Event is the outcome of an experiment
- Examples
- - toss a die observe a 1
- observe an odd number

(1,3,5)

Simple Event an event that cannot be decomposed

is called a simple event Examples -toss a die,

event that observe a 1 is a simple event.

event that observe an odd number is not - How

many simple events are there in the card

experiment? Each experiment will result in

one and only one of the simple events. An

event is a collection of one or more simple

events - the event of observing an odd number is

made up of 3 simple events

- Sample Space is the set of all possible

direct outcomes of an experiment - Example
- - tossing a dice, sample space (1,2,3,4,5,6)
- - tossing two dice?

- Formal Definition of Probability If you repeat

an experiment over and over again say (N) times

and event A is observed n times, then we view

the probability of event as - P(A) n / N
- practically, we cannot get the probability by

using this method. One of approaches we are

going to learn to calculate the probability is

called Simple Events Approach - of simple events included in A
- P(A)
- of simple events included in

sample space. - of possible direct outcomes in A
- P(A)
- of all possible direct outcomes of

the experiment

- Property of simple event probability
- Each probability must lie between 0 and 1,

inclusive. (any number negative or greater than 1

is not probability) - The sum of the probabilities for all simple

events in S equals to 1 (the probability of

sample space is 1)

Example Experiment - toss a coin twice (H head

T tail) Step one list all simple event, table

or tree Step two identify the sample space Step

three calculate probabilities What is the

probability that we observe (H,H)

observe the same type observe the different

type observe at least one head

Problem toss the coin three times. draw tree

diagram how many simple events can we

have? what is sample space? what is prob of

following events observe (H H T) observe

same type three times. observe at most one

head

Learning to Count (all over again)

- Playing the lottery, how many different possible

combinations are there for the winning number - How many different ways can a 8 person batting

order be arranged with 9 players - How many different combinations of cards could be

drawn in Blackjack

How to count when you are picking r things out of

n

- Two questions you have to ask
- Can there be repeats or replacements
- Does the order matter
- Results
- There can be repeats and order matters nr
- There can not be repeats and order matters

permutations - There can not be repeats and order does not

matter combination - There can be repeats and order does matter not

talked about yet

Combination and Permutation Pick r things out of

n at a time, can not have repeats Permutation

formula If position (order) does matter, we have

following number of choices Combination

formula If position (order) does not matter, we

have following number of choices

Intuition of combination and permutation

formula A permutation can be thought of a

selection process in which objects are selected

one by one in a certain order. If we want to

select r objects out of n, the number of first

possible choice is n, and then next is n-1

(because they cannot be the same objects more

than once), and then n-2, finally (n-r1). so the

choices are n(n-1)(n-2) (n-r1) Given one

combination(select r out of n, no matter what

order they are), we should have r! number of

possible permutations. So the number of

combination is number of permutation divided by

r!

- Problem codes edition
- Choose 4 numbers from 09 to comprise a 4-digit

code. - If number cannot be the same, how many codes can

we have? - If number can be the same, how many codes can we

have?

- Problem Cold Stone Ice Cream
- Suppose Cold Stone provide 7 different types of

toppings to put in your ice cream - If you have enough money to choose three

different toppings and the order those toppings

go in does not matter, how many choices do you

have? - If you have enough money to choose 4 different

toppings and the order those toppings go in does

not matter, how many choices do you have?

- Problem Coachs choice
- Ol Roy must choose 5 out of 10 players to put on

his starting lineup. - If players position does not matter, how many

choices the coach has? - If position does matter, how many choices he has?

Event composition and Event relations

Intersection of event A and B, denoted by AB or

( ), is the event that both A and B

occur Example toss a die, what is AB for

following 1) A1, Bobserve odd number

2) Aobserve odd number, Beven number

3) Anumber greater than 4, BSample space

Event Composition and Relations

- Union of event A and B, denoted by AB (

), is the event that A or B or both occur. - Example toss a die, what is AB
- 1) A1, Bobserve odd number
- 2) Aobserve odd number, Beven number
- 3) Anumber greater than 4, BSample space

Event Composition and Relations

- Complement of an event A, denoted by

consists of the all the simple event in the

sample space that are not in A. - Example given following A, what is A complement
- 1) A1
- 2) Aobserve odd number
- 3) Asample space

Event Composition and Relations

- Mutually exclusive/ Disjoint events two

events A and B are said to be mutually exclusive

if when A occurs, B cannot occur(and visa

versa). - This implies that A and B have no common parts

and empty set( ) )

Property of disjoint events

- 1) P(AB) 0
- 2) P( ) P(A) P(B)

S

A

B

- Example toss a die
- Aobserve odd number.
- Bobserve even number
- Are they disjoint events?
- Use simple events approach to calculate P(A),

P(B), P( ) and P( )

- Additive Rule of Probability
- - Given two events A and B,
- P( AB ) P( A ) P( B ) - P( AB )
- If A and B are mutually exclusive, then P(AB)0

and - P(AB) P ( A ) P ( B )

Example toss the coin twice Aobserve at least

one head. Bobserve at least one tail Define

events A, B, AB, and as collection of

simple events. Use simple events approach to

calculate the probability of these events. Use

additive rule of probability to calculate the P(

)

Another important of relationship Given any

event A, P(A) P ( ) 1

Example toss a die twice What is probability of

sum of two observation greater than 3.

Conditional Probability Conditional

probability of event B given A has occurred is

Example toss a die what is probability of

observing a 1 given we already observed an odd

number on the first role

Independence two events A and B are said to

be independent if and only if either P(AB)P(A)

or P(BA)P(B) - The fact that B has occurred

has no effect on the probability that A will

occur

Multiplicative Rule of probability P(AB)P(A)P(B

A)P(B)P(AB) If A and B are independent events,

P(AB)P(A)P(B) Similarly, if A, B and C are

mutually independent events, then

P(ABC)P(A)P(B)P(C)

Problem babys gender a)Two couples plan to have

child this year and what is the probability

of both couples have boys? (Suppose having baby

for each couple are independent events and there

is an equal chance of having a boy or

girl.) b)What is the probability that either

couple or both has a boy? c)If you know one

couple has had a boy what is the probability the

other has had a boy.

How to Solve a Problem

- Read the question and look for the phrase what

it is the probability of/that _____ - Decide how to define the event(s) according to

the blank - Decide if you are looking at an intersection

(and), union (or), conditional (given), or just a

probability - Take the given information and put in in a tree

or a table, then fill in the blanks - Answer the question posed.

Tree

- Multiply up the branches
- Add down the nodes

Probability table for the event A and B there is

a two-way table whose four entries are the four

intersection probabilities P(AB), P(A ), P(

B ), P( ) and whose marginal row and

column sums correspond to the unconditional

probabilities P(A), P( ), P(B) and P( ) as

in the table below B

A P(AB) P(A )

P(A) add across P( B )

P( ) P( ) add across

P(B) P( ) sum down and

across add down add down to

equal 1

Problem Is ABS useful? In a survey involving 100

cars, each vehicle was classifies according to

whether or not it has antilock brakes system(ABS)

and whether or not it has been involved in an

accident in the past year. Suppose that any one

of these cars is randomly selected for

inspection. ABS

No ABS Accident 3

12 No Accident

40 45 Transform this

table into a probability table first.

- Questions
- 1) What is probability that the car has been

involved in an accident in the past year? - 2) What is probability that the car has ABS
- 3) What is probability that the car has not

been in an accident and has ABS - Given that the car has been involved in an

accident, what is the probability that it has

ABS? - Given that the car has ABS, what is the prob that

it has been involved in an accident? - Given that the car does not have ABS, what is the

prob that it has been involved in an accident?

Conditional probability and information

update Consider this example first Suppose 100

students live in Morrison Hall and assume at

first that 50 of them love Chinese food and the

other 50 dont. If a person loves it, the

probability that he order it is 80 for each

meal. If the person doesnt, the probability is

10. Now, we observe an order of Chinese food has

been delivered to the hall. Given this event,

what is the probability the food was ordered by

someone who loves Chinese food?

Bayes Rule Let S1, S2, S3, Sk represent the k

mutually exclusive, only possible states of

nature with prior probabilities P(S1),

P(S2), P(Sk). If an event A occurs, the posterior

probability of Si given A is the conditional

probability

Bayes Rule

- In English The denominator has all possible

ways of getting there while the numerator has the

one way you want - Chinese food example
- P(LO) P(LO)/P(O)
- P(O)P(OL)/(P(O)P(OL)P(O)P(OH))

- Example 2
- One part used in a production process is provided

by three suppliers and supplier 1 provide 20, 2

provide 30 and 3 provide 50. The percentage of

defective parts supplied by these three suppliers

are 0.05, 0.02, and 0.01, respectively. If a part

randomly selected from the production process is

found to be defective, what is the probability

that the part is provided by supplier 1?

Lets Make a Deal

- Monty Hall gives you the choice between 3 doors.

One has a new car behind it and the other 2 have

a goat. Once you have chosen a door Monty,

knowing were the car is, reveals the contents of

one of the doors you have not chosen (always the

goat). He then gives you the option to choose

one of the the other doors. Should you choose

the other door?

- Discrete Random variables and their probability

distribution - Random variable a variable is called a random

variable if the value that it assumes has a

probability of occurring. - 1) Random variable is a quantitative variable
- 2) We can use values from the outcome of an

experiment to get random variables.

Examples of Random Variables

- Toss a die
- xnumber we observe
- xnumber we observe-1
- x0 if observe odd number 1 if

observe even - Flip a coin
- x 0 if H, x 1 if T
- Select a student and measure their height
- x height

Types of Random Variables

- Discrete random variables have a countable

number of values - Continuous random variables have infinite

number of values - Examples
- dice and coin toss discrete
- height - continuous

Probability Distributions

- Probability distribution for a discrete random

variable is a formula, table, or graph that

provides p(x), the probability associated with

each of the value of x. - Example 1 toss a die, xnumber we observe
- formula
- table
- graph
- Example 2 toss two coins, what is the

probability distribution of the the number of

heads that we observe

- Requirements for a discrete probability

distribution - 0
- Sum of P(x)1
- Intuition each simple event is assumed one and

only one value. And each value of x corresponding

to one or more than one simple events, but

different x cannot correspond to same simple

event. So sum of P(x) is sum of simple event

probability.

Expected Value

- Expected value or population mean(mean) of random

variable x with the probability distribution P(x)

is given as - Where the elements are summed over all values of

the random variable x. - Intuitionexpected value is weighted average

value of x

Examples of Expected Values

- If you throw a dice what is the expected value?
- If you flip two coins what is the expected value?
- Should you (if you have rational expectations)

mail (37 cent stamp) a response to Publishers

Clearing House for a chance to win 10,000,000

when the odds of winning are one in 30,000,000?

Variance of a Random Variable

- The variance of random variable x with

probability distribution P(x) and expected value

E(x) is given as - Where the summation is over all values of the

random variable x - The Standard Deviation of random variable x is

equal to the square root of its variance.

Example

- A random variable has the following probability

distribution - Construct a probability histogram
- Find E(x), variance, and standard deviation
- What does Tchebysheff tell us about this

distribution

Example from article in class

- A recent survey showed that 57 of French people

have an unfavorable view of America. You ask at

random 2 French folks what there opinion of

America was, call x the number of French who have

an unfavorable view. - Find the probability distribution and construct

the histogram. - What is the probability that at least one will

dislike American