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Chapter 3 Probability © 2011 Pearson Education, Inc

Contents Events, Sample Spaces, and Probability Unions and Intersections Complementary Events The Additive Rule and Mutually Exclusive Events Conditional Probability The Multiplicative Rule and Independent Events Random Sampling Baye’s Rule As a result of this class, you will be able to ... © 2011 Pearson Education, Inc

Learning Objectives Develop probability as a measure of uncertainty Introduce basic rules for finding probabilities Use probability as a measure of reliability for an inference As a result of this class, you will be able to ... © 2011 Pearson Education, Inc

Thinking Challenge What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean? © 2011 Pearson Education, Inc

Many Repetitions!* Total Heads Number of Tosses 1.00 0.75 0.50 0.25 0.00 25 50 75 100 125 Number of Tosses © 2011 Pearson Education, Inc

Events, Sample Spaces, and Probability
3.1 Events, Sample Spaces, and Probability :1, 1, 3 © 2011 Pearson Education, Inc

Experiments & Sample Spaces
Process of observation that leads to a single outcome that cannot be predicted with certainty Sample point Most basic outcome of an experiment Sample space (S) Collection of all possible outcomes Sample Space Depends on Experimenter! © 2011 Pearson Education, Inc

Sample Space Properties
Mutually Exclusive 2 outcomes can not occur at the same time Male & Female in same person Collectively Exhaustive One outcome in sample space must occur. Male or Female Experiment: Observe Gender © 2011 Pearson Education, Inc © T/Maker Co.

Visualizing Sample Space
1. Listing S = {Head, Tail} 2. Venn Diagram H T S © 2011 Pearson Education, Inc

Sample Space Examples Experiment Sample Space Toss a Coin, Note Face {Head, Tail} Toss 2 Coins, Note Faces {HH, HT, TH, TT} Select 1 Card, Note Kind {2♥, 2♠, ..., A♦} (52) Select 1 Card, Note Color {Red, Black} Play a Football Game {Win, Lose, Tie} Inspect a Part, Note Quality {Defective, Good} Observe Gender {Male, Female} © 2011 Pearson Education, Inc

Events Specific collection of sample points Simple Event Contains only one sample point Compound Event Contains two or more sample points © 2011 Pearson Education, Inc

Venn Diagram Experiment: Toss 2 Coins. Note Faces. Sample Space S = {HH, HT, TH, TT} Compound Event: At least one Tail Other compound events could be formed: Tail on the second toss {HT, TT} At least 1 Head {HH, HT, TH} TH HT Outcome HH TT S © 2011 Pearson Education, Inc

Event Examples Experiment: Toss 2 Coins. Note Faces. Sample Space: HH, HT, TH, TT Event Outcomes in Event Typically, the last event (Heads on Both) is called a simple event. 1 Head & 1 Tail HT, TH Head on 1st Coin HH, HT At Least 1 Head HH, HT, TH Heads on Both HH © 2011 Pearson Education, Inc

Probabilities © 2011 Pearson Education, Inc

What is Probability? 1. Numerical measure of the likelihood that event will cccur P(Event) P(A) Prob(A) 2. Lies between 0 & 1 3. Sum of sample points is 1 1 Certain .5 Impossible © 2011 Pearson Education, Inc

Probability Rules for Sample Points
Let pi represent the probability of sample point i. 1. All sample point probabilities must lie between 0 and 1 (i.e., 0 ≤ pi ≤ 1). 2. The probabilities of all sample points within a sample space must sum to 1 (i.e.,  pi = 1). © 2011 Pearson Education, Inc

Equally Likely Probability
P(Event) = X / T X = Number of outcomes in the event T = Total number of sample points in Sample Space Each of T sample points is equally likely — P(sample point) = 1/T © T/Maker Co. © 2011 Pearson Education, Inc

Steps for Calculating Probability
1. Define the experiment; describe the process used to make an observation and the type of observation that will be recorded 2. List the sample points 3. Assign probabilities to the sample points 4. Determine the collection of sample points contained in the event of interest 5. Sum the sample points probabilities to get the event probability © 2011 Pearson Education, Inc

Combinations Rule A sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible is denoted by and is equal to where the factorial symbol (!) means that For example, 0! is defined to be 1. © 2011 Pearson Education, Inc

Unions and Intersections
3.2 Unions and Intersections :1, 1, 3 © 2011 Pearson Education, Inc

Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways. © 2011 Pearson Education, Inc

Unions & Intersections
Outcomes in either events A or B or both ‘OR’ statement Denoted by  symbol (i.e., A  B) 2. Intersection Outcomes in both events A and B ‘AND’ statement Denoted by  symbol (i.e., A  B) © 2011 Pearson Education, Inc

Event Union: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit. Ace Black Event Black: 2, 2,..., A Sample Space: 2, 2, 2, ..., A S Event Ace: A, A, A, A Event Ace  Black: A, ..., A, 2, ..., K © 2011 Pearson Education, Inc

Event Union: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Simple Event Ace: A, A, A, A Sample Space (S): 2, 2, 2, ..., A Type Red Black Total Ace Ace & Ace & Ace Red Black Non-Ace Non & Non & Non- Red Black Ace Total Red Black S Event Ace  Black: A,..., A, 2, ..., K Simple Event Black: 2, ..., A © 2011 Pearson Education, Inc

Event Intersection: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit. Ace Black Event Black: 2,...,A Sample Space: 2, 2, 2, ..., A S Event Ace: A, A, A, A Event Ace  Black: A, A © 2011 Pearson Education, Inc

Event Intersection: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Simple Event Ace: A, A, A, A Sample Space (S): 2, 2, 2, ..., A Type Red Black Total Ace Ace & Ace & Ace Red Black Non-Ace Non & Non & Non- Red Black Ace Event Ace  Black: A, A Total Red Black S Simple Event Black: 2, ..., A © 2011 Pearson Education, Inc

Compound Event Probability
1. Numerical measure of likelihood that compound event will occur 2. Can often use two–way table Two variables only © 2011 Pearson Education, Inc

Event Probability Using Two–Way Table
Total 1 2 A P(A B ) P(A B ) P(A ) 1 1 1 1 2 1 A P(A B ) P(A B ) P(A ) 2 2 1 2 2 2 Total P(B ) P(B ) 1 1 2 Joint Probability Marginal (Simple) Probability © 2011 Pearson Education, Inc

Two–Way Table Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2/52 2/52 4/52 Non-Ace 24/52 24/52 48/52 P(Ace) Total 26/52 26/52 52/52 P(Red) P(Ace  Red) © 2011 Pearson Education, Inc

Thinking Challenge P(A) = P(D) = P(C  B) = P(A  D) = P(B  D) = What’s the Probability? Event C D Total A 4 2 6 B 1 3 5 10 Let students solve first. Allow about 20 minutes for this. © 2011 Pearson Education, Inc

Solution* The Probabilities Are: P(A) = 6/10 P(D) = 5/10 P(C  B) = 1/10 P(A  D) = 9/10 P(B  D) = 3/10 Event C D Total A 4 2 6 B 1 3 5 10 © 2011 Pearson Education, Inc

3.3 Complementary Events :1, 1, 3 © 2011 Pearson Education, Inc

Complementary Events Complement of Event A The event that A does not occur All events not in A Denote complement of A by AC S AC A © 2011 Pearson Education, Inc

Rule of Complements The sum of the probabilities of complementary events equals 1: P(A) + P(AC) = 1 S AC A © 2011 Pearson Education, Inc

Complement of Event Example
Experiment: Draw 1 Card. Note Color. Black Sample Space: 2, 2, 2, ..., A S Event Black: 2, 2, ..., A Complement of Event Black, BlackC: 2, 2, ..., A, A © 2011 Pearson Education, Inc

The Additive Rule and Mutually Exclusive Events
3.4 The Additive Rule and Mutually Exclusive Events :1, 1, 3 © 2011 Pearson Education, Inc

Mutually Exclusive Events
Events do not occur simultaneously A  B does not contain any sample points © 2011 Pearson Education, Inc

Mutually Exclusive Events Example
Experiment: Draw 1 Card. Note Kind & Suit. Outcomes in Event Heart: 2, 3, 4 , ..., A Sample Space: 2, 2, 2, ..., A Mutually Exclusive What is the intersection of mutually exclusive events? The null set. S Event Spade: 2, 3, 4, ..., A Events  and are Mutually Exclusive © 2011 Pearson Education, Inc

Additive Rule Used to get compound probabilities for union of events P(A OR B) = P(A  B) = P(A) + P(B) – P(A  B) For mutually exclusive events: P(A OR B) = P(A  B) = P(A) + P(B) © 2011 Pearson Education, Inc

Additive Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2 4 Non-Ace 24 48 26 52 Try other examples using this table. P(Ace  Black) = P(Ace) + P(Black) P(Ace Black) = – = © 2011 Pearson Education, Inc

Thinking Challenge Using the additive rule, what is the probability? P(A  D) = P(B  C) = Event C D Total A 4 2 6 B 1 3 5 10 Let students solve first. Allow about 10 minutes for this. © 2011 Pearson Education, Inc

Solution* Using the additive rule, the probabilities are: 1. P(A  D) = P(A) + P(D) – P(A  D) = – = 2. P(B  C) = P(B) + P(C) – P(B  C) = – = © 2011 Pearson Education, Inc

Conditional Probability
3.5 Conditional Probability :1, 1, 3 © 2011 Pearson Education, Inc

Conditional Probability
1. Event probability given that another event occurred 2. Revise original sample space to account for new information Eliminates certain outcomes 3. P(A | B) = P(A and B) = P(A  B) P(B) P(B) © 2011 Pearson Education, Inc

Conditional Probability Using Venn Diagram
Black ‘Happens’: Eliminates All Other Outcomes Ace Black Black S (S) Event (Ace  Black) © 2011 Pearson Education, Inc

Conditional Probability Using Two–Way Table
Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2 4 Non-Ace 24 48 26 52 Revised Sample Space Try other examples using this table. © 2011 Pearson Education, Inc

Thinking Challenge Using the table then the formula, what’s the probability? P(A|D) = P(C|B) = Event C D Total A 4 2 6 B 1 3 5 10 Let students solve first. Allow about 20 minutes for this. © 2011 Pearson Education, Inc

Solution* Using the formula, the probabilities are: © 2011 Pearson Education, Inc

The Multiplicative Rule and Independent Events
3.6 The Multiplicative Rule and Independent Events :1, 1, 3 © 2011 Pearson Education, Inc

Multiplicative Rule 1. Used to get compound probabilities for intersection of events 2. P(A and B) = P(A  B) = P(A)  P(B|A) = P(B)  P(A|B) 3. For Independent Events: P(A and B) = P(A  B) = P(A)  P(B) © 2011 Pearson Education, Inc

Multiplicative Rule Example
Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2 2 4 Try other examples using this table. Non-Ace 24 24 48 Total 26 26 52 P(Ace  Black) = P(Ace)∙P(Black | Ace) © 2011 Pearson Education, Inc

Statistical Independence
1. Event occurrence does not affect probability of another event Toss 1 coin twice 2. Causality not implied 3. Tests for independence P(A | B) = P(A) P(B | A) = P(B) P(A  B) = P(A)  P(B) © 2011 Pearson Education, Inc

Thinking Challenge Using the multiplicative rule, what’s the probability? Event C D Total A 4 2 6 B 1 3 5 10 P(C  B) = P(B  D) = P(A  B) = Let students solve first. Allow about 10 minutes for this. © 2011 Pearson Education, Inc

Solution* Using the multiplicative rule, the probabilities are: © 2011 Pearson Education, Inc

Tree Diagram Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace. Dependent! R P(R  R)=(6/20)(5/19) =3/38 5/19 R 6/20 14/19 B P(R  B)=(6/20)(14/19) =21/95 R 6/19 P(B  R)=(14/20)(6/19) =21/95 14/20 B 13/19 B P(B  B)=(14/20)(13/19) =91/190 © 2011 Pearson Education, Inc

3.7 Random Sampling :1, 1, 3 © 2011 Pearson Education, Inc

Importance of Selection
How a sample is selected from a population is of vital importance in statistical inference because the probability of an observed sample will be used to infer the characteristics of the sampled population. © 2011 Pearson Education, Inc

Random Sample If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a random sample. © 2011 Pearson Education, Inc

Random Number Generators
Most researchers rely on random number generators to automatically generate the random sample. Random number generators are available in table form, and they are built into most statistical software packages. © 2011 Pearson Education, Inc

3.8 Bayes’s Rule :1, 1, 3 © 2011 Pearson Education, Inc

Bayes’s Rule Given k mutually exclusive and exhaustive events B1, B1, Bk , such that P(B1) + P(B2) + … + P(Bk) = 1, and an observed event A, then © 2011 Pearson Education, Inc

Bayes’s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are defective, while 1% of Factory II’s are defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I? © 2011 Pearson Education, Inc

Bayes’s Rule Example Defective 0.02 Factory I 0 .6 0.98 Good Defective 0.01 0 .4 Factory II 0.99 Good © 2011 Pearson Education, Inc

Key Ideas Probability Rules for k Sample Points, S1, S2, S3, , Sk ≤ P(Si) ≤ 1 2. © 2011 Pearson Education, Inc

Key Ideas Random Sample All possible such samples have equal probability of being selected. © 2011 Pearson Education, Inc