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QUANTITATIVE ANALYSIS

QUANTITATIVE ANALYSIS. UNIT 4: Probability. TouchText. Classical Probability Theory Law of Large Numbers Compound Events and Venn Diagrams Intersections, Unions and Mutually Exclusive Events Conditional Probability and Independent Events. Problems and Exercises. Next.

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QUANTITATIVE ANALYSIS

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  1. QUANTITATIVE ANALYSIS UNIT 4: Probability TouchText • Classical Probability Theory • Law of Large Numbers • Compound Events and Venn Diagrams • Intersections, Unions and Mutually Exclusive Events • Conditional Probability and Independent Events Problems and Exercises Next

  2. Random Experiments, Events and Sample Spaces Random Experiments are those whose outcomes are determined by chance (luck). Outcomes of Random Experiments are also called “Events”. All possible events taken together constitute the experiment’s Sample Space. Dictionary Outcome 1 = Event 1 Outcome 2 = Event 2 Outcome 3 = Event 3 Outcome 4 = Event 4 Etc. Random Experiment Sample Space Event = what does, in fact, happen. Sample Space = everything that might happen. Back Next

  3. Venn Diagram Venn Diagrams show events inside a sample space. Dictionary Not A = Ac Sample Space A The Complement of an event is it NOT occurring. Complement A = “Not A” = Ac Back Next

  4. Compound Events Compound Experiments are the combination of two or more experiments. Compound Events are the combination of two or more events. Dictionary + Coin Toss Roll of Dice ≥ *Actually, in English, it is one “die” (singular) and two or more “dice” (plural). But we use “dice” to avoid confusion. Back Next

  5. Compound Events on a Venn Diagram Both events by themselves must be inside the sample space. When there are only two (2) events, A and B … SampleSpace Event A Event B Dictionary With two compound events, there are only 4 possible outcomes … 1. A and B 2. A not B 3. B not A 4. Not A Not B Not A Not B Sample Space A And B B not A A not B Back Next

  6. Venn Diagram: The Intersection of A and B The “Intersection” of two events is when they both occur. The intersection of events A AND B is denoted Dictionary Not A and Not B Sample Space B not A A not B Back Next

  7. Mutually Exclusive Events Exception to previous result: Mutually Exclusive events do not have an intersection: If one is true, the other event cannot be true. Dictionary With Mutually Exclusive events, there are only 3 possible outcomes … 1. A not B 2. B not A 3. Not A Not B B. From Viet Nam A. From Thailand Not from Thailand or Viet Nam (Example) Sample Space: What country is someone from? Back Next

  8. Venn Diagram: The Union of A and B The “Union” of two events is when either occurs. The union of events A OR B is denoted Dictionary Not A and Not B Sample Space Back Next

  9. Example: The Union of A and B Example: A university wants to give a job to a “mature” teacher with either (a) 5+ years experience, OR (b) at least 35 years old. Dictionary < 35 years old and < 5 years work experience Sample Space ≥ 35 years of age 5+ years experience Back Next

  10. Probabilities: Random Events, Outcomes and Sample Spaces Random experiments are those whose outcomes are determined by chance (luck). All possible outcomes constitutes the experiment’s Sample Space. Eachpossible outcome (“event”) has a known, objective probability(Pr) of occurring. … Dictionary …Thus, the probability of getting an outcome in the sample space is one. Event 1 Event 2 Event 3 Event 4 Etc. Experiment Pr(1) Pr(2) Pr(3) Pr(4) Probabilities (Pr) Pr(Sample Space) = 1 Sample Space Back Next

  11. Probabilities For all random experiments … Dictionary Sample Space Heads (front of coin) Pr(Heads) = 0.5 Pr(Tails) = 0.5 Tails (back of coin) Event = Coin Toss Back Next

  12. Classical Probability When all events are equally likely to occur (e.g. coin toss, roll of dice, babies, etc.) then … Dictionary Pr(five) = 1/6 Pr(heads) = 1/2 Back Next

  13. Law of Large Numbers The Law of Large Numbers states that as the number of experimental trials increases, the observed frequency of an event approaches its classical probability. Dictionary Heads = 1; Tails = 0 Back Next

  14. Probabilities: Events and Complements By definition, Event A and its complement “Not A” (AC) have probabilities that add to 1. Example: Event A = student is Cambodian in a school with 1,000 total students, of which 85 are Cambodian. If a student is chosen at random …. Dictionary Sample Space Pr(SS) = 1 (1,000 students) Not A = Ac AC = not Cambodian (915) A = Cambodian (85) Pr(Cambodian) = 85/1,000 = 8.5% Pr(Not Cambodian) = 915/1,000 = 91.5% 8.5% + 91.5% = 100% Pr(A)+ Pr(AC) = 1. Back Next

  15. Probabilities: Compound Events When there are only two (2) events A and B Pr(Sample Space) = 1 ….. So …… Dictionary 4 3 1 2 With two compound events, there are only 4 possible outcomes … 1. A and B 2. A not B 3. B not A 4. Not A Not B Not A and Not B 1 Sample Space 3 2 4 B not A A not B Back Next

  16. Union of Events When there are two compound events, the probability of their union (either A OR B) can be written as: Dictionary 4 Sample Space 4 B not A A not B Not A and Not B * You have to subtract in the equation because it is part of both Pr(A) and Pr(B). So otherwise it would be counted twice. 4 Back Next

  17. Example: Intersection and Union of Events A particular ASEAN school has the following break-down of where students are from, and what they study. Dictionary 815 80 100 5 A = Cambodian B = Math Student Back Next

  18. Probabilities: Union of Mutually Exclusive Events When two compound events are Mutually Exclusive, the probability of their intersection (both A AND B) is 0. Thus, the probability of their union can be written as: Dictionary Sample Space A B Not A and Not B Back Next

  19. Probability of Mutually Exclusive Events Dictionary B. From Viet Nam (40) From Thailand (305) Not from Thailand or Viet Nam (655) (Example) Sample Space: What country is a student from? Back Next

  20. Conditional Probability A Conditional Probability is the probability of one event occurring, given another event has already occurred. Dictionary Example: What is the probability that it will rain today, given that it rained yesterday? The symbol “|” is used to mean “given” what is to the right of the symbol is known and/or true. Back Next

  21. Conditional Probability For the last 50 years, on July 1 weather has been recorded in Bangkok and Singapore. Question: What is the probability that (A) it is raining in Bangkok, given that (B) it is raining in Singapore? Dictionary Event A = Rain Bangkok 1 7 2 No Rain (40) Event B = Rain Singapore Back Next

  22. Conditional Probability: The Other Way For the last 50 years, on July 1 weather has been recorded in Bangkok and Singapore. Question: What is the probability that (B) it is raining in Singapore, given that (A) it is raining in Bangkok? Dictionary Event A = Rain Bangkok 1 7 2 No Rain (40) Event B = Rain Singapore Back Next

  23. Alternative View of Conditional Probabilities: Restricting the Sample Space One way to think about a conditional probability is that it limits your sample space, and thus changes the probabilities. Dictionary Pr(A|B) B is true. B is now the sample space, with Pr(B = 100%). Original Sample Space B true (A also true) New Sample Space = B A true B true A true B true Not A and Not B Back Next

  24. Comparing Conditional Distributions With Conditional Probabilities For the last 50 years, on July 1 weather has been recorded in Bangkok and Singapore. Question: What is the probability that (A) it is raining in Singapore, given that (B) it is raining in Bangkok? Dictionary Event A = Rain Bangkok 1 7 2 No Rain (40) Event B = Rain Singapore Back Next

  25. Alternative View of Compound Events: Tree Diagrams Tree Diagrams are another useful way of showing compound events. Assuming A is known before B …. Dictionary Pr(B|A) A B B ? Pr(A) A ? ) Pr(BC|A) A B Pr(B|AC) A B Pr(Ac) B ? Pr(BC|AC) A B Sample Space Back Next

  26. Alternative View of Conditional Probabilities: Tree Diagrams Tree Diagrams are another useful way of showing conditional probabilities. Assuming A is known before B …. Dictionary Pr(B|A) A B B ? Pr(A) A ? ) Pr(BC|A) A B Pr(B|AC) A B Pr(Ac) B ? Pr(BC|AC) A B Sample Space * Compare conditional probability formula with top branch of the tree diagram: Pr(A)*Pr(B|A) = Back Next

  27. Previous Example: Tree Diagrams Dictionary Pr(B|A) = 80/85 B ? Pr(A) = 85/1000 Pr(BC|A) = 5/85 A ? Pr(B|AC) = 100/915 Pr(Ac) = 915/1000 B ? A = Cambodian AC = Not Cambodian B = Study Math BC = Don’t Study Math Pr(BC|AC) = 815/915 Back Next

  28. Example: Cars and Motos Venn Diagram Example: Out of 100 students sampled … Moto only = 72 Car only = 15 Both Car and Moto = 10 No Car or Moto = 3 Dictionary Moto 3 Car 72 10 15 Back Next

  29. Example: Cars Motos Venn Diagram What is the probability of a student having either a car OR a motorbike? Dictionary = 82 + 25 – 10 = 97 (and 3 with nothing) Moto 3 Car 72 10 15 Back Next

  30. Example: Car|Moto Venn Diagram Example: Out of 100 students sampled … What is the probability of a student having a car GIVEN that the student has a motorbike? Moto only = 72 Car only = 15 Both Car and Moto = 10 No Car or Moto = 3 Total = 100 Students Dictionary Moto 3 Car 72 10 15 (= 72 + 10) Back Next

  31. Example: Car Moto Venn Diagram Moto only = 72 Car only = 15 Both Car and Moto = 10 No Car or Moto = 3 Total = 100 Students Dictionary Pr(B|A) = 10/82 B ? Pr(A) = 82/100 Pr(BC|A) = 72/82 A ? Pr(B|AC) = 15/18 Pr(Ac) = 18/100 B ? A = Moto AC = No Moto B = Car BC = No Car Pr(BC|AC) = 3/18 Back Next

  32. Independent Events When events are Independent, knowing one event provides no additional information in assessing the probability of another event. Dictionary Knowing the gender (boy/girl) of the 1st child does not change the probabilities for the gender of the 2nd child. Pr(B|B) = 0.5 B = boy; G = girl Pr(B) = 0.5 Pr(G|B) = 0.5 Pr(B|G) = 0.5 Pr(G) = 0.5 Event A = 1st child Event B = 2nd child Pr(G|G) = 0.5 Back Next

  33. Conditional Distributions (categorical data) Revisited Example: Reconsider example from previous unit (Categorical Data) . Dictionary 65.4% = 364/557 25.3% = 141/557 9.3% = 52/557 100% = 557/557 For example, for the Bangkok campus, the (conditional) distribution of students across courses is: Back Next

  34. Independent Events Events are considered Independent Events when knowing the outcome of one event provides no additional information in providing probabilities of the outcome of another event. Dictionary Example: Pr(Male|Car) = 13/552 = ¼ Pr(Male) = 23/92 = 1/4 Back Next

  35. Summary of Equations Compound Events (* A and B occur at the same time) Dictionary Conditional Probability (* one event (B) occurs before the other event (A)) This equation can be re-written as: Back Next

  36. Summary of Equations, Part 2 Compound Events Dictionary + Conditional Probability = combined formulas (by substitution) or Back Next

  37. End of Unit 4 Questions and Problems Directions: (a) complete the three other probabilities in this sample space. (b) Create a similar tree diagram with B as the first event. Moto only = 72 Car only = 15 Both Car and Moto = 10 No Car or Moto = 3 Total = 100 Dictionary Pr(B|A) = 10/82 B ? Pr(A) = 82/100 ? Pr(BC|A) = 72/82 A ? ? Pr(B|AC) = 15/18 Pr(Ac) = 18/100 B ? ? A = Moto AC = No Moto B = Car BC = No Car Pr(BC|AC) = 3/18 Back End

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