Chapter 5 Probability in Our Daily Lives

1. Chapter 5Probability in Our Daily Lives • Learn …. About probability – the way we quantify uncertainty How to measure the chances of the possible outcomes of random phenomena How to find and interpret probabilities

2. Section 5.1 How Can Probability Quantify Randomness?

3. Randomness • Applies to the outcomes of a response variable • Possible outcomes are known, but it is uncertain which will occur for any given observation

4. Some Popular Randomizers • Rolling dice • Spinning a wheel • Flipping a coin • Drawing cards

5. Random Phenomena • Individual outcomes are unpredictable • With a large number of observations, predictable patterns occur

6. Random Phenomena • With random phenomena, the proportion of times that something happens is highly random and variable in the short run but very predictable in the long run.

7. Jacob Bernoulli: Law of Large Numbers • As the number of trials of a random phenomenon increases, the proportion of occurrences of any given outcome approaches a particular number “in the long run”.

8. Probability • With a random phenomenon, the probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations.

9. Roll a Die What is the probability of rolling a ‘6’? • .22 • .10 • .17

10. Question about Random Phenomena • If a family has four girls in a row and is expecting another child, does the next child have more than a ½ chance of being a boy?

11. Independent Trials • Different trials of a random phenomenon are independent if the outcome of any one trial is not affected by the outcome of any other trial.

12. Section 5.2 How Can We Find Probabilities?

13. Sample Space • For a random phenomenon, the sample space is the set of all possible outcomes

14. Example: Roll a Die Once • The Sample Space consists of six possible outcomes: {1, 2, 3, 4, 5, 6}

15. Example: Flip a Coin Twice • The Sample Space consists of the four possible outcomes: {(H,H) (H,T) (T,H) (T,T)}

16. Example: A 3-Question Multiple Choice Quiz • Diagram of the Sample Space

17. Tree Diagram • An ideal way of visualizing sample spaces with a small number of outcomes • As the number of trials or the number of possible outcomes on each trial increase, the tree diagram becomes impractical

18. Event • An event is a subset of the sample space

19. Probabilities for a Sample Space • The probability of each individual outcome is between 0 and 1 • The total of all the individual probabilities equals 1

20. Example: Assigning Subjects to Echinacea or Placebo for Treating Colds Experiment • Multi-center randomized experiment to compare an herbal remedy to a placebo for treating the common cold • Half of the volunteers are randomly chosen to receive the herbal remedy and the other half will receive the placebo • Clinic in Madison, Wisconsin has four volunteers • Two men: Jamal and Ken • Two women: Linda and Mary

21. Example: Assigning Subjects to Echinacea or Placebo for Treating Colds • Sample Space to receive the herbal remedy: {(Jamal, Ken), (Jamal, Linda), (Jamal, Mary), (Ken, Linda), (Ken, Mary), (Linda, Mary)} • These six possible outcomes are equally likely

22. Example: Assigning Subjects to Echinacea or Placebo for Treating Colds • What is the probability of the event that the sample chosen to receive the herbal remedy consists of one man and one woman?

23. Probability of an Event • The probability of an event A, denoted by P(A), is obtained by adding the probabilities of the individual outcomes in the event. • When all the possible outcomes are equally likely:

24. Example: What are the Chances of a Taxpayer being Audited? • Each year, the Internal Revenue Service audits a sample of tax forms to verify their accuracy

25. Example: What are the Chances of a Taxpayer being Audited?

26. Example: What are the Chances of a Taxpayer being Audited? • What is the sample space for selecting a taxpayer? {(under $25,000, Yes), (under $25,000, No), ($25,000 - $49,000, Yes) …}

27. Example: What are the Chances of a Taxpayer being Audited? • For a randomly selected taxpayer in 2002, what is the probability of an audit?

28. Example: What are the Chances of a Taxpayer being Audited? • For a randomly selected taxpayer in 2002, what is the probability of an income of $100,000 or more?

29. Basic Rules for Finding Probabilities about a Pair of Events • Complement of an Event • Intersection of 2 Events • Union of 2 Events

30. Complement of an Event Complement of Event A: (see next slide) • Consists of all outcomes in the sample space that are not in A • Is denoted by Ac • The probabilities of A and Ac add to 1 • P(Ac) = 1 – P(A)

31. Complement of an Event

32. Disjoint Events • Two events, A and B, are disjoint if they do not have any common outcomes

33. Example: Disjoint Events • Pop Quiz: 3 Multiple-Choice Questions • Event A: Student answers exactly 1 question correctly • Event B: Student answer exactly 2 questions correctly

34. Example: Disjoint Events

35. Intersection of Two Events • The intersection of A and B: consists of outcomes that are in both A and B

36. Union of Two Events • The union of A and B: Consists of outcomes that are in A or B • In probability, “A or B” denotes that A occurs or B occurs or both occur

37. Intersection and Union of Two Events

38. How Can We Find the Probability that A or B Occurs? Addition Rule: Probability of the Union of Two Events • For the union of two events, P(A or B) = P(A) + P(B) – P(A and B) • If the events are disjoint, P(A and B) = 0, so P(A or B) = P(A) + P(B)

39. How Can We Find the Probability that A and B Occurs? • Multiplication Rule: Probability of the Intersection of Independent Events • For the intersection of two independent events, A and B: P(A and B) = P(A) x P(B)

40. Example: Two Rolls of A Die • P(6 on roll 1 and 6 on roll 2): 1/6 x 1/6 = 1/36

41. Example: Guessing on a Pop Quiz • Pop Quiz with 3 Multiple-choice questions • Each question has 5 options • A student is totally unprepared and randomly guesses the answer to each question

42. Example: Guessing on a Pop Quiz • The probability of selecting the correct answer by guessing = 0.20 • Responses on each question are independent

43. Tree Diagram for the Pop Quiz

44. Example: Guessing on a Pop Quiz • What is the probability that a student answers at least 2 questions correctly? P(CCC) + P(CCI) + P(CIC) + P(ICC) = 0.008 + 3(0.032) = 0.104

45. Events Often Are Not Independent • Example: A Pop Quiz with 2 Multiple Choice Questions • Data giving the proportions for the actual responses of students in a class Outcome: II IC CI CC Probability: 0.26 0.11 0.05 0.58

46. Events Often Are Not Independent • Define the events A and B as follows: • A: {first question is answered correctly} • B: {second question is answered correctly}

47. Events Often Are Not Independent • P(A) = P{(CI), (CC)} = 0.05 + 0.58 = 0.63 • P(B) = P{(IC), (CC)} = 0.11 + 0.58 = 0.69 • P(A and B) = P{(CC)} = 0.58 • If A and B were independent, P(A and B) = P(A) x P(B) = 0.63 x 0.69 = 0.43

48. Question of Independence • Don’t assume that events are independent unless you have given this assumption careful thought and it seems plausible

49. Example: A family has two children If each child is equally likely to be a girl or boy, find the probability that the family has two girls. • 1/2 • 1/3 • 1/4 • 1/8

50. Section 5.4 (just for fun) Applying the Probability Rules