Probability

10/6/09

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 1

Probability

Statistics 111 - Lecture 8

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 2

Administrative Notes

• Homework 2 due in recitation: Friday, Oct. 9

• Homework 3 now posted on course website:

http://stat.wharton.upenn.edu/~stjensen/stat111.html

• Recitation is cancelled for Friday, Oct. 16th

• Homework 3 must be submitted to your TAs

mailbox (Huntsman Hall 4th floor) by noon on

Friday, Oct 16th

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 3

Why do we need Probability?

• We have several graphical and numerical

statistics for summarizing our data

• We want to make probability statements

about the significance of our statistics

• Eg. In our class, mean(height) = 66.7 inches

• What is the chance that the true height of Penn

students is between 60 and 70 inches?

• Eg. r = -0.22 for draft order and birthday

• What is the chance that the true correlation is

significantly different from zero?

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 4

Deterministic vs. Random Processes

• In deterministic processes, the outcome can

be predicted exactly in advance

• Eg. Force = mass x acceleration. If we are given

values for mass and acceleration, we exactly know

the value of force

• In random processes, the outcome is not

known exactly, but we can still describe the

probability distribution of possible outcomes

• Eg. 10 coin tosses: we don’t know exactly how

many heads we will get, but we can calculate the

probability of getting a certain number of heads

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 5

Events

• An event is an outcome or a set of outcomes of

a random process

Example: Tossing a coin three times

Event A = getting two heads = {HTH, HHT, THH}

Example: Picking real number X between 1 and 20

Event A = chosen number is over 8.23 = {X ≤ 8.23}

Example: Tossing a fair dice

Event A = result is an even number = {2, 4, 6}

• Notation: P(A) = Probability of event A

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 6

Combinations of Events

• The complement Ac of an event A is the event that A

does not occur

• Complement Rule : P(Ac) = 1 - P(A)

• The union of two events A and B is the event that

either A or B or both occurs

• The intersection of two events A and B is the event

that both A and B occur

Event A Complement of A Union of A and B Intersection of A and B

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Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 1

Statistics 111 - Lecture 8

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 2

Administrative Notes

Homework 2 due in recitation: Friday, Oct. 9
Homework 3 now posted on course website: http://stat.wharton.upenn.edu/~stjensen/stat111.html
Recitation is cancelled for Friday, Oct. 16th
Homework 3 must be submitted to your TAs mailbox (Huntsman Hall 4th^ floor) by noon on Friday, Oct 16th Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 3

Why do we need Probability?

We have several graphical and numerical statistics for summarizing our data
We want to make probability statements about the significance of our statistics
Eg. In our class, mean(height) = 66.7 inches
- What is the chance that the true height of Penn students is between 60 and 70 inches?
Eg. r = -0.22 for draft order and birthday
- What is the chance that the true correlation is significantly different from zero? Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 4

Deterministic vs. Random Processes

In deterministic processes, the outcome can be predicted exactly in advance - Eg. Force = mass x acceleration. If we are given values for mass and acceleration, we exactly know the value of force
In random processes, the outcome is not known exactly, but we can still describe the probability distribution of possible outcomes - Eg. 10 coin tosses: we don’t know exactly how many heads we will get, but we can calculate the probability of getting a certain number of heads Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 5

Events

An event is an outcome or a set of outcomes of a random process Example: Tossing a coin three times Event A = getting two heads = {HTH, HHT, THH} Example: Picking real number X between 1 and 20 Event A = chosen number is over 8.23 = {X ≤ 8.23} Example: Tossing a fair dice Event A = result is an even number = {2, 4, 6}
Notation: P(A) = Probability of event A Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 6

Combinations of Events

The complement Ac^ of an event A is the event that A does not occur
Complement Rule : P(Ac) = 1 - P(A)
The union of two events A and B is the event that either A or B or both occurs
The intersection of two events A and B is the event that both A and B occur Event A Complement of A Union of A and B Intersection of A and B

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 7 Disjoint Events

Two events are called disjoint if they can not happen at the same time - Events A and B are disjoint means that the intersection of A and B is zero
Example: coin is tossed twice
- S = {HH,TH,HT,TT}
- Events A={HH} and B={TT} are disjoint
Disjoint Rule: If A and B are disjoint events then P(A or B) = P(A) + P(B) Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 8 Independent events - Events A and B are independent if knowing that A occurs does not affect the probability that B occurs - Example: tossing two coins Event A = first coin is a head Event B = second coin is a head - Disjoint events cannot be independent! - If A and B can not occur together (disjoint), then knowing that A occurs does change probability that B occurs - Multiplication Rule: If A and B are independent P(A and B) = P(A) x P(B) Independent Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 9

Example: Independence in Use

How Many Of Me? www.howmanyofme.com “There are 305 , 244 ,385 people in the United States of America…How many people share your name?” Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 10

Example: Independence in Use

How did they find come up with this estimate of 59 people named “Shane Jensen”?
If you assume independence of first and last name, and let N = number of people in US
Is independence a reasonable assumption here? €

(Shane Jensen) = N × P(Shane Jensen)

= N × P(Shane) × P(Jensen) = N × 144991 N

× 125150

N

€ Assuming Independence! Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 11

Combining Probability Rules Together

Initial screening for HIV in the blood first uses an enzyme immunoassay test (EIA)
Even if an individual is HIV-negative, EIA has probability of 0.006 of giving a positive result
Suppose 100 people are tested who are all HIV-negative. What is probability that at least one will show positive on the test?
First, use complement rule: P(at least one positive) = 1 - P(all negative) Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 12
Now, we assume that each individual is independent and use the multiplication rule for independent events:
P(test negative) = 1 - P(test positive) = 0.
So, we finally we have € P(all negative) = P(test 1 negative) × × P(test 100 negative) € P(all negative) = 0.994 × × 0.994 = (0.994)^100 € P(at least one positive) = 1 − (0.994)^100 = 0.

Combining Probability Rules Together

Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 19

Continuous Random Variables

Continuous random variables have a non- countable number of values
Can’t list the entire probability distribution, so we use a density curve instead of a histogram
Eg. Normal density curve: Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 20

Calculating Continuous Probabilities

Discrete case: add up bars from probability histogram
Continuous case: we have to use integration to calculate the area under the density curve:
Although it seems more complicated, it is often easier to integrate than add up discrete “bars” - If a discrete r.v. has many possible values, we often treat that variable as continuous instead Oct. 6, 2009 Stat 111 - Lecture 8 - Probability 21 Next Class – Lecture 9
Probability Distributions
Moore and McCabe: Section 4.3,1.

Probability - Introductory Statistics - Lecture Slides | STAT 111, Study notes of Statistics

Related documents

Partial preview of the text

Download Probability - Introductory Statistics - Lecture Slides | STAT 111 and more Study notes Statistics in PDF only on Docsity!

Probability

Statistics 111 - Lecture 8

Administrative Notes

Why do we need Probability?

Deterministic vs. Random Processes

Events

Combinations of Events

Example: Independence in Use

Example: Independence in Use

(Shane Jensen) = N × P(Shane Jensen)

× 125150

N

N

Combining Probability Rules Together

Combining Probability Rules Together

Continuous Random Variables

Calculating Continuous Probabilities