Statistics and Modelling Course

1. Statistics and Modelling Course 2011

2. Topic: Introduction to Probability Part of Achievement Standard 90643 Solve straightforward problems involving probability 4 Credits Externally Assessed NuLake Pages 102  130

3. Lesson 1: Sample space and combined events • Sample space. • Complementary events. • The intersection between 2 events. • The union between 2 events. Use new edition of Sigma (photocopy): Do Sigma Ex. 6.02 (p105) – Venn Diagrams. Ex. 6.03 (p108) Q2 & 3 – Contingency Tables

4. Basic Probability Sample space : set of all possible outcomes of an experiment. Event: any subset of sample space. E.g. A video game involves a player having 3 shots at a target. If he gets 2 or more hits, he gets an extra bonus shot, and if after that he has 3 or more hits, he gets a final shot. • Show the possible outcomes on a tree diagram. • List the sample space.

5. Population: Year 13 SAM class at STCDRAW ON BOARD ALONGSIDE PROJECTOR IMAGE

6. Venn Diagrams – represent events within a Sample Space (s) Sample Space – the set of all possible outcomes. All possible outcomes – probabilities add to 1 (i.e. 100%). Does NOT take calc. P(C’) = 1- p(C) = = Takes Calc. P(C) =

7. Complementary Events (notes) For any event “A” A: Event A occurs. A’: The Complement of A. Event A does NOT occur. where P(A’) = 1 – P(A) Example: If we select a person at random from this class, the events: “This class member takes Calculus” , C and “This class member does NOT take Calculus”, C` are complements of each other. P( C’) = 1 – P(C) = 1 – =

8. Takes Physics (Ph) Takes Calc (C) THE INTERSECTION“A” AND “B”

9. THE INTERSECTION“A” AND “B” Takes Physics (Ph) Takes Calc (C)

10. THE INTERSECTION“A” AND “B” Takes Calc AND Physics (C  Ph) P(C  Ph) =

11. THE UNION“A” OR “B” (or both) P(C  Ph) = Takes Calc OR Physics (or both) (C  Ph)

12. THE UNION“A” OR “B” (or both) P(C)

13. P(Ph) P(C) THE UNION“A” OR “B” (or both) +

14. THE UNION“A” OR “B” (or both) – P(C  Ph)

15. THE UNION“A” OR “B” (or both) So for any 2 events A and B: P(A  B) = P(A) + P(B) – P(A  B) Takes Calc OR Physics (or both) (C  Ph) E.g. to find the union of a student chosen at random from this class taking either Calc or Physics (or both), we add the individual probabilities of the events ‘takes Calc’ and ‘takes Physics’, then subtract the intersection (the overlap).

16. NOTES: For any 2 events A and B : The intersection – BOTH AandB happen P(AB) = P(A) + P(B) – P(AB) E.g. The probability that a randomly chosen member of this class takes both calculus and physics. The union – A orB happen (or both): P(AB) = P(A) + P(B) – P(AB) E.g. to find the union of a student chosen at random from this class taking either Calc or Physics (or both), we add the individual probabilities of the events ‘takes Calc’ and ‘takes Physics’, then subtract the intersection (the overlap), so that we don’t count it twice.

17. Venn diagram Do Sigma pg. 6 – Ex. 1.2 (old version) pg. 105 – Ex. 6.02 (new version) *HW: Handout (contingency tables)

18. Lesson 3: Mutually exclusive & independent events • Mutually exclusive events. • Independence. 1. Notes on mutual exclusivity & independence. 2. Do Sigma (old) – Ex. 1.3 (pg. 9) Or Sigma (new) – Ex. 6.04 (pg. 111) 3. HW: Probability assignment (due Thurs.)

19. Mutually Exclusive Events

20. Takes Geography(G) Mutually Exclusive Events Takes Physics (Ph) There is NO OVERLAP. If one happens, the other can’t! If someone takes Geo, then he can’t take Physics. If someone takes Physics, then he can’t take Geo. So P(Geo  Ph) =?

21. Takes Geography(G) Mutually Exclusive Events Takes Physics (Ph) There is NO OVERLAP. If one happens, the other can’t! If someone takes Geo, then he can’t take Physics. If someone takes Physics, then he can’t take Geo. So P(Geo  Ph) =0

22. Mutually Exclusive Events (notes) If two events are Mutually Exclusive, it means that if one happens, the other cannot. So they cannot both occur. At STC in Year 13, you cannot take both Geography and Physics. The two events “Takes 13Geo” and “Takes 13Physics” are Mutually Exclusive. Test whether 2 events are mutually exclusive by finding whether or not there is any intersection between them P(AB). Can they both occur? 2 events A and B are mutually exclusive if and only if P(A B) = 0

23. Independence

24. Independence If two events are Independent, it means that the event that one has occurred does NOT alter the probability of the other occurring. Examples of two events that are independent: Examples of two events that are not independent:

25. How to test for independence: • Any 2 events A and B are independent if and only if: P(AB) = P(A) × P(B) • If P(AB) ≠ P(A) × P(B), then events A and B are NOT Independent. Work – finish for HW: 1. Do Sigma pg. 9 – Ex. 1.3 (old version) pg. 111 – Ex. 6.04 (new version).

26. More combined events examples Do worksheet of past NCEA qs (involves unions, intersections, mutually exclusive events and the ‘neither’ event – using Venn Diagrams and Contingency Tables).

27. Lesson 4: Tree diagrams 1 • Use tree diagrams to calculate probabilities. STARTER: The “Neither” event. Do NuLake: Tree Diagrams • Pg. 115120 - Q1626(h) Complete for HW.

28. The ‘Neither’ event

29. 2006 NCEA exam question Rewa asked 150 randomly chosen students what programmes they watched last night. 90 watched Shortland Street, 50 NZ Idol, and 30 had watched both. What is the probability that a randomly chosen student had watched neither Shortland Street nor NZ Idol last night?

30. Next: Tree Diagrams – Do NuLake pg. 115 120

31. The ‘NEITHER’ event P(A`∩B`) P(A υ B)

32. The ‘NEITHER’ event P(A`∩B`) P(A υ B) “Nor” = 1- “Or” P(Neither A nor B) = 1 – P(A υ B) This can be easily observed on a contingency table (see previous example). P(Neither A nor B) is written as P(A`∩ B`), the probability that event A doesn’t occur AND event B doesn’t occur.

33. Lesson 5: Tree diagrams 2 • Practice and develop more confidence with using Tree Diagrams to solve problems involving probability. Do Sigma pg. 12 – Ex. 1.4 (old version): Q1-14 or, in new edition, pg. 116 – Ex .6.05 * Extension people: Do new edition – pg. 116 – Ex. 6.05: Do Q314, 15(Exc), then read infinite probabilities example at bottom of p115, then do Q16-20 (all Exc).

34. Lesson 6: Conditional probability 1. Intro toConditional Probability • Introduction to Conditional Probability and its notation and formula. • Link to Tree Diagrams. Do NuLake pg. 123126 Or Sigma Ex. 6.01, 6.02

35. Conditional Probability Conditional probability means the probability of a particular event occurring GIVEN that some other event has occurred. E.g. the probability that a randomly chosen person in this class has a car, GIVEN that we know he has a job. Once we know that a particular event has occurred our sample space of possibilities is reduced.

36. Conditional Probability NUMBER IN CLASS TODAY = ____ Q. Stand up if you have a job. n(Job) = P(Job) = Q. Of those standing, RAISE YOUR HAND if you ALSO own a car. n(JobCar) = P(JobCar) =

37. Conditional Probability NUMBER IN CLASS TODAY = ____ Q. Stand up if you have a job. n(Job) = P(Job) = Q. Of those standing, RAISE YOUR HAND if you ALSO own a car. n(JobCar) = P(JobCar) = The probability that a randomly selected member of this class has a car GIVEN that he has a job is given by: P(CarᅵJob) = Number standing up with hands upTotal number standing

38. Conditional Probability NUMBER IN CLASS TODAY = ____ Q. Stand up if you have a job. n(Job) = P(Job) = Q. Of those standing, RAISE YOUR HAND if you ALSO own a car. n(JobCar) = P(JobCar) = The probability that a randomly selected member of this class has a car GIVEN that he has a job is given by: P(CarᅵJob) = Number standing up with hands up = n( ? ) Total number standing n( ? )

39. Conditional Probability NUMBER IN CLASS TODAY = ____ Q. Stand up if you have a job. n(Job) = P(Job) = Q. Of those standing, RAISE YOUR HAND if you ALSO own a car. n(JobCar) = P(JobCar) = The probability that a randomly selected member of this class has a car GIVEN that he has a job is given by: P(CarᅵJob) = Number standing up with hands up = n(Job Car) Total number standing n(Job) HAVE A GO: Do NuLake pg. 123 - JUST Question 29. Show this information on a contingency table.

40. Conditional Probability on a Venn diagram P(A’B’) P(AB`) P(BA`) P(AB)

41. Conditional Probability on a Venn diagram P(A’) P(AB`) P(AB)

42. Conditional Probability on a Venn diagram We KNOW Event A has occurred

43. Conditional Probability on a Venn diagram We KNOW Event A has occurred P(BᅵA)

44. Conditional Probability on a Venn diagram HAVE A GO: Do NuLake pg. 124  126 You MUST do: Q30-40. * Extension: Q41. RULE: The conditional probability that event B occurs if we already know that A has happened is written as P(BᅵA). We say “the probability of B, given A”. We KNOW Event A has occurred P(BᅵA) P(BᅵA) = P(AB) P(A) Something to ponder: If A and B are independent, then P(BᅵA) = P(___)

45. Lesson 7: Conditional probability 2. • Watch youtube clip on conditional probability: http://www.youtube.com/watch?v=4PwnvqGEHoU • Link with tree-diagrams. Look at one together. Which probabilities on the diagram are conditional probabilities? • Practice solving problems involving conditional probability using Venn diagrams. • Go over HW qs from NuLake & finish these off. 2. Faster people move on to NuLake pages 130134.

46. Lesson 8 (if time): Conditional probability 3. • Problem-solving: The Base Rate Fellacy. • Last lesson to practice conditional probability problems and sort out any difficulties. 1.) Finish NuLake pages 130134 2.) Sigma (new): Pg. 176 – Ex. 9.04 + answer qs for Unlawful Activity investigation. Extension: 2010 NCEA exam (AS90643) – final question (hard Venn diagram problem).

47. The base-rate fellacy

48. The base-rate fellacy In a certain town there are two types of taxis; 90% are Blue Cabsand 10% are Red Cabs.

49. The base-rate fellacy In a certain town there are two types of taxis; 90% are Blue Cabsand 10% are Red Cabs. A taxi is involved in a hit-and-run accident.

50. The base-rate fellacy In a certain town there are two types of taxis; 90% are Blue Cabsand 10% are Red Cabs. A taxi is involved in a hit-and-run accident. One witness says it was a Red Cab. The court tests this witness’s ability to distinguish between Blueand Redcabs under identical conditions and concludes that the witness is right 80% of the time. Question: What is the probability that the taxi was a Red Cab? Lawyers, judges, doctors and even maths teachers have claimed that the answer is 80%. Yet this is incorrect.