Understanding Probability Theory: Foundations and Rules

1. Stat 155, Section 2, Last Time • Probability Theory • Foundations of Probability • Events, Sample Space • Probability Function • Simple Random Sampling (count samples) • Big Rules of Probability: • Not Rule ( 1 – P{opposite}) • Or Rule

2. Reading In Textbook Approximate Reading for Today’s Material: Pages 259-271 , 311-323 Approximate Reading for Next Class: Pages 277-286, 291-305

3. Midterm I Coming up: Tuesday, Feb. 27 Material: HW Assignments 1 – 6 Extra Office Hours: Mon. Feb. 26, 8:30 – 12:00, 2:00 – 3:30 (Instead of Review Session) Bring Along: 1 8.5” x 11” sheet of paper with formulas

4. Midterm I How will I test for Excel skills? • No computers allowed • Fill out menus with pencil • Write Excel Commands with pencil Put Excel commands (& details) on your: 1 8.5” x 11” sheet of paper with formulas

5. Midterm I Example: What fraction of N(1,2) population is smaller than 0? Could ask you to fill out menu: You write: 0 1 2 true

6. Midterm I Example: What fraction of N(1,2) population is smaller than 0? Above results in: Note: “command line”

7. Midterm I Example: What fraction of N(1,2) population is smaller than 0? So could ask you to simply write: =NORMDIST(0,1,2,TRUE) Note that you need to know: • Excel function names • Which arguments go where So put all these on your sheet of formulas (suggestion: make early & use to study)

8. Big Rules of Probability • Not Rule: P{not A} = 1 – P{A} • Or Rule: P{A or B} = P{A} + P{B} – P{A and B} Third rule? Symbolic logic is based on: and, or, not How about a rule for and?

9. Big Rules of Probability • Now head towards a rule for “and” • Needs a new concept: Conditional Probability Idea: If event A is known to have occurred, what is chance of B? Note: “knowing A” means sample space is restricted to A

10. Conditional Probability E.g. Roll a die, A = {even}, B = {1,2,3} P{B, when A is known} = ??? (i.e. Somebody rolls, and only tells you “even”. Note “<= 3” is no longer 50-50, Since fewer even #s are <= 3)

11. Conditional Probability E.g. Roll a die, A = {even}, B = {1,2,3} P{B, when A is known} = ??? Try “equally likely”: CAREFUL: This is wrong!!! Problem: for B, should not include 1 or 3, since they are not even

12. Conditional Probability E.g. Roll a die, A = {even}, B = {1,2,3} P{B, when A is known} = ??? Correct Answer: Makes sense, since chance should go down from ½.

13. Conditional Probability General definition: Probability of B given A = Next, by multiplying by P{A}, get and rule of probability

14. And Rule of Probability Big Rule III: P{A & B} = P{A|B} P{B} = P{B|A} P{A} Memory trick: like “canceling fractions”, but make bar vertical, not a fraction Note: 2 ways to do this. Good strategy: look at both, as one is often easier.

15. The And Rule of Probability HW: 4.89, 4.91a 4.95 (see 4.92)

16. And now for something completely different Recall Distribution of majors of students in this course:

17. And now for something completely different Three nurses died & went to heaven where they were met at the Pearly Gates by St. Peter.

18. And now for something completely different To the first, he asked, "What did you do on Earth and why should you go to heaven?" "I was a nurse in an inner city hospital," she replied. "I worked to bring healing and peace to the poor suffering city children." "Very noble," said St. Peter. "You may enter." And in through the gates she went.

19. And now for something completely different To the next, he asked the same question, "So what did you do on Earth?" "I was a nurse at a missionary hospital in Africa," she replied. "For many years, I worked with a skeleton crew of doctors and nurses who tried to reach out to as many peoples and tribes with a hand of healing and with a message of God's love." "How touching," said St. Peter. "You too may enter." And in she went.

20. And now for something completely different He then came to the last nurse, to whom he asked, "So, what did you do back on Earth?" After some hesitation, she explained, "I was just a nurse at an H.M.O." St. Peter pondered this for a moment, and then said, "Okay, you may enter also." "Whew!" said the nurse. "For a moment there, I thought you weren't going to let me in."

21. And now for something completely different "Oh, you can come in," said St. Peter, "but you can only stay for three days..."

22. Big Rules of Probability Example illustrating power (and use) of rules: Toss a Coin: if H take a ball from I: R RG G G if T take a ball from II: R RG Now study progressively harder problems…

23. Balls in Urns Example H  R RG G G T  R RG E.g. A: P{R | H} = 2/5 (chance of R, if know got H) Simple “equally likely” calculation (just counting) works here

24. Related HW HW: C13 A company makes 40% of its cars at factory A, and the rest at factory B. Factory A produces 10% lemons, and Factory B produces 5% lemons. A car is chosen at random. What is the probability that: • It came from Factory A? (0.4) • It is a lemon, if it came from Fact. A? (0.1)

25. Balls in Urns Example H  R RG G G T  R RG E.g. B: P{R & H} = ??? Try simple counting: P{R & H} = ??? Caution: This is wrong!!! Reason: balls are not equally likely.

26. Balls in Urns Example H  R RG G G T  R RG E.g. B: P{R & H} = ??? Correct Answer: P{R & H} = P{H | R} P{R} (OK, but hard) = P{R | H} P{H} = (2/5)(1/2) = 1/5 Note: < ¼ (from wrong answer above)

27. Related HW HW: C13 • It is a lemon, from Factory A? (0.04) (think carefully about contrast with (b))

28. Balls in Urns Example H  R RG G G T  R RG E.g. C: P{R} = ??? Try simple counting: P{R} = ??? Caution: This is wrong!!! Reason: again balls are not equally likely.

29. Balls in Urns Example H  R RG G G T  R RG E.g. C: P{R} = ??? Note: now expect > ½, since R’s in II are more likely (thus get more weight) Need to take which urn into account, so write event in terms of the urn ball came from

30. Balls in Urns Example H  R RG G G T  R RG E.g. C: Correct Answer: P{R} = P{(R & H) or (R & T)} = (“expand”) = P{R & H} + P{R & T} – 0 (or Rule) = 1/5 + P{R | T} P{T} = (from B) = 1/5 + (2/3)(1/2) = 8/15 Note: slightly > ½ (as expected)

31. Related HW HW: C13 • It is a lemon? (0.07)

32. Balls in Urns Example H  R RG G G T  R RG E.g. D: P{H | R} = ??? • Saved for last, since this is hardest • Although only “turn around” of e.g. A • This is common: One Cond. Prob. much easier than the reverse

33. Balls in Urns Example H  R RG G G T  R RG E.g. D: P{H | R} = Makes sense: if see R, less likely from H

34. Related HW HW: C13 • It came from Factory A, if it is a lemon? (4/7) 4.103 4.105

35. Plotting Bivariate Data Recall Toy Example: (1,2) (3,1) (-1,0) (2,-1)

36. And now for something completely different Viewing Higher Dimensional Data: • Extend to higher dimensions • E.g. replace pairs by triples • Make “3-d scatterplot” • As “points in space” • Think about “point cloud”

37. And now for something completely different Toy 3-d data set:

38. And now for something completely different High Light One Point

39. And now for something completely different X Coor of High Light

40. And now for something completely different Y Coor of High Light

41. And now for something completely different Z Coor of High Light

42. And now for something completely different Proj- ection on X Axis

43. And now for something completely different 1-d View: Proj- ection on X Axis

44. And now for something completely different Proj- ection on Y Axis

45. And now for something completely different 1-d View: Proj- ection on Y Axis

46. And now for something completely different Proj- ection on Z Axis

47. And now for something completely different 1-d View: Proj- ection on Z Axis

48. And now for something completely different Proj- ection on X-Y Plane

49. And now for something completely different Proj- ection on X-Y Plane rotated up

50. And now for something completely different Proj- ection on X-Z Plane