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5: CHAPTER PROBABILITY 5.1 What is probability? Random experiment results in one of a number of possible outcomes. The outcome that occurs cannot be predicted with certainty. Examples: Tossing a coin, rolling a die. Sample Space (S) - the list of all possible outcomes of a random experiment Experiment Tossing a coin once Rolling a die once Tossing a coin twice H=Head, T=Tail Example: Tossing a coin twice (tree diagrams) 1st toss H 2nd toss Outcomes H T HH HT Sample Space S = {H, T} S = {1,2,3,4,5,6} S = {HH, HT, TH, TT} T H T 1 TH TT Event - any subset of a sample space Sample Space S Event A Examples: Experiment Tossing a coin Rolling a die Rolling a coin twice S Event {H, T} Head is obtained = {H} {1,2,3,4,5,6} Number of dots is even ={2,4,6} {HH, HT, The same outcome in both TH, TT} tosses ={HH, TT} P(A) -the probability of an event A 1. Relative frequency definition of probability: P(A) = the fraction of time the event A occurs in a long series of repetitions of a random experiment. Example: Experiment = tossing a fair coin, event A = head is obtained. Then P(A) = 0.50. 2 2. Classical definition of probability: If S consists of m equally likely outcomes, an event A consists of k outcomes in, then P( A) = k m Example: Experiment: A fair die is rolled once. Event A: even number of dots is obtained. What is P(A)? Example: Experiment: A fair die is rolled twice. Event A: Total number of dots facing up exceeds 10. What is P(A)? Example: A fair coin is tossed twice. Find the probability that at least 1 head occurs. 3 5.2 Venn Diagrams S = the interior of a rectangle, outcomes in S are points in the rectangle, events are regions inside the rectangle S A Ac -A does not occur (all outcomes in S that are not in A) A Ac A or B - union of A and B (all outcomes that are either in A or B or both) A B A or B (at least one of A and B occurs) 4 A and B - intersection of A and B (all outcomes that are in both A and B) A A and B B A and B (A and B both occur) A and B are disjoint if the have no outcomes in common (can never occur simultaneously) A B Disjoint events Example: A survey of 200 graduates provided the following data: Working (W) 80 40 Not Working 40 40 Males (M) Females (F) Find the probabilities that a randomly selected graduate is a working person, a male, a working male. 5 5.3 Probability Rules 1. 2. A any event, then 0 P(A) 1, P(S) =1, S A P(A) the is probability that a randomly selected point in S belongs to the region A because Area ( A) P ( A) P( A) = = = P ( A). Area ( S ) P ( S ) 1 3. 4. P(Ac) = 1 - P(A), Addition Rule: If A and B are any two events, then P(A or B) = P(A) + P(B) - P(A and B). A B If A and B are disjoint, then P(A or B) = P(A) + P(B). 6 Example: Refer to the data from the survey. What is the probability that a randomly selected person is a male or working? Example: You have torn a tendon and are facing surgery to repair it. Infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occurs together in 1%. What percent of the operations succeed and are free from infection? Example: What is the probability of winning the main jackpot in 6-49 game? Solution: 7 5.4 Conditional Probability Example (Survey continued): (a) Find the probability that a randomly selected person works? Find the probability that the person works if the person is a man? (b) P(B|A) - the probability of an event B given that the event A has occurred B A P(B|A) = the size of the common part A and B relative to that of A. The conditional probability P(B|A) of an event B given an event A is defined by P ( B | A) = P ( A _ and _ B ) P ( A) when P(A) >0. 8 Example: Use the formula to recalculate the probability (b) in the above example. Conclusion: P(A and B) = P(A) P(B|A) Tree Diagram Interpretation: Event P(A) A P(B|A) B A and B P(A) P(B|A) = P(A and B) Example: If we randomly pick up two television tubes from a shipment of 240 TV tubes of which 15 are defective, what is the probability that (a) both are nondefective, (b) at least one of the two is nondefective. Consider sampling without and with replacement. 9 Solution: 5.5 Independence Two events A and B are independent if P(B|A) = P(B). Collorary: The two events A and B are independent if and only if P(A and B) = P(A) P(B). 10 Example: A fair coin is tossed, and then a fair die is rolled. What is the probability of obtaining a head and a 6? Example: The primary cooling unit in a nuclear plant has reliability (the probability that the unit will not fail) 0.999. There is also a back-up cooling unit to substitute for the primary unit when it fails. The reliability of the back-up is 0.890. Find the reliability of the cooling system of the power plant. Assume that the primary and backup units work independently. Henryk Kolacz September 2007 11
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Path: New Mexico >> CHNE >> 525 Fall, 2008
Description: Series Solutions to Linear Ordinary Differential Equations I Power Series Solution for the Harmonic Oscillator Equation : ODE: Transform independent variable Transform derivatives d2y dx 2 2 y 0 x dy dy d dx d dx d2y dx 2 d2y dx 2 dy d d dx 2 ...
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Path: Brandeis >> MATH >> 15a Spring, 2008
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Path: Brandeis >> MATH >> 15a Spring, 2008
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Path: Brandeis >> COSI >> 30a Spring, 2008
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Path: Brandeis >> MATH >> 15a Spring, 2008
Description: Otto Bretscher Linear Algebra with Applications 3rd Edition Section 2.3 4. 6. 16. 20. ...
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Path: Brandeis >> MATH >> 15a Spring, 2008
Description: Otto Bretscher Linear Algebra with Applications 3rd Edition Section 1.2 2. 4. 6. 8. ...
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Path: Brandeis >> MATH >> 15a Spring, 2008
Description: Otto Bretscher Linear Algebra with Applications 3rd Edition Section 3.3 2. Redundant vectors: Basis of image: Basis of kernel: 4. Redundant vectors: none Basis of image: Basis of kernel: 6. Redundant vectors: Basis of image: Basis of kernel: 8. R...
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Path: Brandeis >> MATH >> 15a Spring, 2008
Description: Otto Bretscher Linear Algebra with Applications 3rd Edition Section 6.1 10. By Sarrus\'s rule, is invertible. 24. is not invertible when , which is when . 32. By Fact 6.1.6, 44. For an matrix , : For any : For any matrix . , , , so . , so Th...
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Path: New Mexico >> CHNE >> 525 Fall, 2008
Description: Series Solutions to Linear Ordinary Differential Equations III Examples: Frobenius\' Solution about Regular Singular Points Text Example 2, pp. 253, 3rd ed Text Example 2, pp. xxx, 2nd ed ODE: 3xy\' \' y\' y 0 1 1 Standard form: y\' \' y\' y 0 3x 3x 1 Px 3x...
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Path: New Mexico >> CHNE >> 525 Fall, 2008
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Path: University of Alberta >> STAT >> 151 Fall, 2007
Description: CHAPTER 1: INTRODUCTION 1.1 What is Statistics? Questions to explore: 1. What is the population of Canada? What is the population of Alberta? Canada: 31,612,897, Alberta: 3,306, 359 (2006). Census (every member of the population counted). Also data c...
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Path: New Mexico >> CHNE >> 524 Fall, 2008
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Path: New Mexico >> CHNE >> 525 Fall, 2008
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Path: New Mexico >> CHNE >> 524 Fall, 2008
Description: NE-524 Interaction of radiation with Matter Problem Set #4 (Complete by 11/6/2005) 1) Give 1 gram of At-218 (T1/2 = 1.5 seconds), and assuming that all the alphas (E=6.694 MeV) interact with an Aluminum absorber in 0.1 seconds, how much energy is dep...
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Path: New Mexico >> CHNE >> 524 Fall, 2008
Description: NE-524 Interaction of radiation with Matter Problem Set #5 (Complete by 11/13/2005) 1) A beam of 0.52 MeV electrons pass into a large reservoir of He at 3 atm pressure, 20 oC (electrons are totally absorbed). a) What is the range of the electrons? b)...
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Path: New Mexico >> CHNE >> 515 Fall, 2006
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Path: New Mexico >> CHNE >> 515 Fall, 2006
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Path: New Mexico >> CHNE >> 515 Fall, 2006
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Path: New Mexico >> CHNE >> 515 Fall, 2006
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Path: New Mexico >> CHNE >> 515 Fall, 2006
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Path: Maryland >> ENES >> 102 Spring, 2008
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Path: UCSB >> HIST >> 80 Spring, 2008
Description: History 80: East Asian Civilization Spring Quarter 2008 T-TH 9:30-10:45, Buchanan Hall, 1910 Sections as assigned. Instructor: Anthony Barbieri-Low HSSB 4225 805-893-4065 (no msg.) barbieri-low@history.ucsb.edu Office Hours: Tues. 12:30-2:30 TA\'s: ...
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Path: Maryland >> ENES >> 102 Spring, 2008
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Path: Maryland >> ENES >> 102 Spring, 2008
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Path: Maryland >> ENES >> 102 Spring, 2008
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Path: Maryland >> ENES >> 102 Spring, 2008
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Path: Syracuse >> CSD >> 315/615 Spring, 2008
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Path: Old Dominion >> ECI >> 301 Fall, 2007
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Path: UCSB >> HIST >> 80 Spring, 2008
Description: History 80: East Asian Civilization Spring 2008 Essay no. 1: DUE IN LECTURE: TUESDAY, APRIL 22, 2008 Your first essay will be an exercise in creative historical fiction, drawing from the lecture on the Warring States\' philosophers and from the passag...
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Path: CUNY City >> ECONOMICS >> 290 Spring, 2007
Description: Principles of Statistics First Examination Fall 2007 Name: Date: Instructions: Answer ALL the questions in the blue book. There is no need to write an explanation for the true or false questions or for the multiple-choice questions. For the problems...
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: Syracuse >> CSD >> 315/615 Spring, 2008
Description: Anatomy Of Respiration Respiration Respiration Inspiration Inhalation Breathing in Brings oxygen to the body\'s cells Expiration Exhalation Breathing out Eliminates waste products 1 Boyle\'s Law Pressure =force distributed over area P=F/A An incre...
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Path: CUNY City >> ECONOMICS >> 290 Spring, 2007
Description: Principles of Statistics Second Examination Fall 2007 Name: Date: Instructions: Answer ALL the questions in the blue book. There is no need to write an explanation for the true or false questions or for the multiple-choice questions. For the problem...
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: CUNY City >> ECONOMICS >> 290 Spring, 2007
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: CUNY City >> ECONOMICS >> 290 Spring, 2007
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Path: CUNY City >> CHEM >> 103 Spring, 2007
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: Texas >> BIO >> 325 Spring, 2008
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Path: Bucknell >> EDUC >> 243 Winter, 2008
Description: Urban Anthro. STUDY GUIDE 2 Urbanism- UL (61-63) Definitions of City How does a person define the concept of a city? Where exactly is the line drawn between city, suburb, and rural areas? Personal reference is a subjective way to define a city. Where...
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Path: Bucknell >> EDUC >> 243 Winter, 2008
Description: Learning Log 2 Knowledge It is necessary to understand how children grow and develop to realize how to effectively teach, and in the second chapter of the text, the Piaget, Vygotsky, and Erikson views of cognitive, personal and social development are...
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Path: Bucknell >> EDUC >> 243 Winter, 2008
Description: Learning Log 10 Knowledge The main idea of chapter 10 is that motivation, which is an internal process that activates, guides, and maintains behavior over time, is critically important for students and teachers in the classroom. Theories of motivatio...
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Path: Bucknell >> EDUC >> 243 Winter, 2008
Description: Learning Log: Chapter 3 Knowledge The main idea is that children develop physically through changes in muscle development at early ages and later in life through puberty, mentally by becoming fully literate, socially with peer interaction and cogniti...
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Path: Bucknell >> ANTHRO >> 290 Winter, 2008
Description: URBAN ANTHROPOLOGY-STUDY GUIDE EXAM 1 Urban Danger: Life in a Neighborhood of Strangers Sally Engle Merry pp.115-126 Ethnographic study of a multiethnic housing project in a high-crime neighborhood shows how the boundaries between social groups cont...
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Path: UCSB >> ECE >> 162c Spring, 2008
Description: ECE 162C: PROBLEM SET #2 DUE WEDNESDAY, APRIL 16, 2008 PROBLEMS: 1. Derive an expression for the confinement factor of single mode fibers defined as the fraction of the total mode power contained inside the core . Use the Gaussian approximation for...
HW_Set_13_1
Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
Description: ...
CSDL3
Path: Syracuse >> CSD >> 315/615 Spring, 2008
Description: Anatomy Of Respiration Structures of Respiration Bony thorax Visceral thorax Muscles 1 The Rib Cage Forms the front, sides, and back of the bony thorax Mobility Consists of 12 thoracic vertebrae Sternum 12 pairs of ribs The Rib Cage 2 The Ribs ...
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Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
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Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
Description: ...
HW_Set_14_2
Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
Description: ...
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Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal Entry #1 1. Having a separate justice system for juveniles can be traced back all the way to common law in England. They stated that children under seven should not face legal punishments. However, in the...
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Path: Syracuse >> CSD >> 315/615 Spring, 2008
Description: Anatomy of Respiration Muscles of Expiration Muscles of Expiration Thorax Abdomen Passive Expiration Back Forced Expiration 1 Internal Intercostal Interosseous Portion O: Inferior margin of ribs 1-11 I: Superior surface of the rib below In: Interc...
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Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal 9 1. The reintegration philosophy, used by community-based programs, states that the community that the offender will be reentered into also must be changed; not just the offender. The community is just a...
jjjournal8
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal 8 1. Probation is the suspension of a jail sentence. A juvenile who has been convicted of a crime does not serve jail time. The court orders probation and the juvenile returns to the community for a certa...
jjjournal5
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal 5 1. Preventive detention occurs after the juvenile is taken by the police to the intake personnel. The intake personnel then review the case and decide if the juvenile should be taken to juvenile court. ...
jjjournal6
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Professor Khondaker Juvenile Justice System Journal 6 1. A guardian ad litem is \"usually a lawyer who is appointed by the court to take care of youths who need help, especially in neglect, dependency, and abuse cases, but also occasionally in delinqu...
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Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
Description: ...
introlit
Path: Kutztown >> ENG >> 010 Fall, 2007
Description: Professor Casner English 010-090 Female Narrator in \"A Rose for Emily\" \"A Rose for Emily\" by William Faulkner is a tragic story. It is a tale of a woman who cannot accept the death of her father of her lover Homer. She stays secluded from her town. T...
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Path: LSU Shreveport >> CHEM >> 121 Fall, 2008
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Path: Kutztown >> CRJ >> 101 Spring, 2008
Description: Chapter 6 The difference between screening in vs. screening out applicants for police positions. o Screening in process of identifying police applicants who are the best-qualified candidate for the applicant pool o Screening out process identifying...
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Path: Kutztown >> CRJ >> 101 Spring, 2008
Description: Chapter 1 The distinction between civil & criminal law o Civil law is laws concerned with relationships between individuals (contracts, business transactions, family relations) o Criminal law is laws concerned with the relationship between the indivi...
jjjournal3
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal Entry 3 1. Jeremy Bentham and Cesare Beccaria are the creators of the idea of Classical criminology. Bentham and Beccaria \"viewed humans as rational creatures who were willing to surrender enough liberty ...
jjjournal4
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal 4 1. There are nine factors that influence the disposition of juveniles by police. The first is the nature of the offense. If the offense is more serious, the more likely the police will arrest the juveni...
jjjournal1
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Dr. Khondaker Juvenile Justice System Journal Entry #1 1. Having a separate justice system for juveniles can be traced back all the way to common law in England. They stated that children under seven should not face legal punishments. However, in the...
sgp1
Path: Kutztown >> CRJ >> 221 Spring, 2008
Description: Chapter 1 The distinction between civil & criminal law o Civil law is laws concerned with relationships between individuals (contracts, business transactions, family relations) o Criminal law is laws concerned with the relationship between the indivi...
CSDL5
Path: Syracuse >> CSD >> 315/615 Spring, 2008
Description: Physiology of Respiration Respiration Inspiration Quiet Forced Expiration Passive Forces of torque, elasticity, and gravity Active Muscular activity Passive Expiration Torque Chondral portion of the rib cage Inspiration Torquing strain Elastic c...
