1. CS 573: Advanced AI Probabilities & Monte-Hall Problem

2. CS 573 : Advanced AIProbabilities & Monte-Hall Problem2 In a perfect simple world …  Actions have deterministic effects  States are completely observable

3. CS 573 : Advanced AIProbabilities & Monte-Hall Problem3 But, this world is not perfect … For example, a robot in the field  Actions -> uncertain effects  Observations -> noises & errors  Unpredictable events Rocks fall from sky Robots gain intelligence and rebel Uncertainties => Probabilities

4. CS 573 : Advanced AIProbabilities & Monte-Hall Problem4 Notation P(VAR = value) The probability variable VAR takes on the given value P(STUDENTS=20) = 0.93 P(Propositional_sentence) P(Propositional_sentence=true) The probability the given propositional sentence is true P(HAPPY) = 0.40 P(HAPPY AND HEALTHY) = 0.39

5. CS 573 : Advanced AIProbabilities & Monte-Hall Problem5 Conditional probabilities Rainy 0.36 No Stars 0.44 Rainy & No Stars 0.30

6. CS 573 : Advanced AIProbabilities & Monte-Hall Problem6 Conditional probabilities Rainy 0.36 No Stars 0.44 Rainy & No Stars 0.30 We know its rainy today, what ’ s the probability that last night there is no stars in sky?

7. CS 573 : Advanced AIProbabilities & Monte-Hall Problem7 Conditional probabilities Rainy 0.36 No Stars 0.44 Rainy & No Stars 0.30 P(No Stars | Rainy)

8. CS 573 : Advanced AIProbabilities & Monte-Hall Problem8 Conditional probabilities Rainy 0.36 No Stars 0.44 Rainy & No Stars 0.30 P(No Stars | Rainy) = P(Rainy & No Starts) / P(Rainy)

9. CS 573 : Advanced AIProbabilities & Monte-Hall Problem9 Conditional probabilities Rainy 0.36 No Stars 0.44 Rainy & No Stars 0.30 P(No Stars | Rainy) = P(Rainy & No Starts) / P(Rainy) = 0.30 / 0.36

10. CS 573 : Advanced AIProbabilities & Monte-Hall Problem10 Conditional probabilities P(A | B) = P(A & B) / P(B)

11. CS 573 : Advanced AIProbabilities & Monte-Hall Problem11 Axioms of probabilities 1.0 <=P(A) <= 1 2.P(true) = 1; P(false) = 0 3.P(A or B) = P(A) + P(B) – P(A & B) A B A & B

12. CS 573 : Advanced AIProbabilities & Monte-Hall Problem12 Example derivation P(A) + P(NOT A) = 1

13. CS 573 : Advanced AIProbabilities & Monte-Hall Problem13 Example derivation P(A) + P(NOT A) = 1 P(A or B) = P(A) + P(B) – P(A & B)

14. CS 573 : Advanced AIProbabilities & Monte-Hall Problem14 Example derivation P(A) + P(NOT A) = 1 P(A or B) = P(A) + P(B) – P(A & B)

15. CS 573 : Advanced AIProbabilities & Monte-Hall Problem15 Example derivation P(A) + P(NOT A) = 1 P(A or B) = P(A) + P(B) – P(A & B) P(A or NOT A) = P(A) + P(NOT A) – P(A & NOT A) Let B be NOT A

16. CS 573 : Advanced AIProbabilities & Monte-Hall Problem16 Example derivation P(A) + P(NOT A) = 1 P(A or B) = P(A) + P(B) – P(A & B) P(A or NOT A) = P(A) + P(NOT A) – P(A & NOT A) Let B be NOT A P(true) P(false)

17. CS 573 : Advanced AIProbabilities & Monte-Hall Problem17 Example derivation P(A) + P(NOT A) = 1 P(A or B) = P(A) + P(B) – P(A & B) P(A or NOT A) = P(A) + P(NOT A) – P(A & NOT A) Let B be NOT A P(true) P(false) 1 = P(A) + P(NOT A) – 0

18. CS 573 : Advanced AIProbabilities & Monte-Hall Problem18 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 Happy Healthy

19. CS 573 : Advanced AIProbabilities & Monte-Hall Problem19 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 Happy Healthy

20. CS 573 : Advanced AIProbabilities & Monte-Hall Problem20 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 Happy Healthy

21. CS 573 : Advanced AIProbabilities & Monte-Hall Problem21 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 Happy Healthy

22. CS 573 : Advanced AIProbabilities & Monte-Hall Problem22 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 Happy Healthy

23. CS 573 : Advanced AIProbabilities & Monte-Hall Problem23 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 P(NOT Healthy)

24. CS 573 : Advanced AIProbabilities & Monte-Hall Problem24 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 P(NOT Healthy) = P(Healthy=false)

25. CS 573 : Advanced AIProbabilities & Monte-Hall Problem25 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 P(NOT Healthy) = P(Healthy=false) =P(Happy & NOT Healthy) + P(NOT Happy & NOT Healthy)

26. CS 573 : Advanced AIProbabilities & Monte-Hall Problem26 Joint probability distribution Happy Healthy P true true 0.39 true false 0.01 false true 0.06 false false 0.54 P(NOT Healthy) = P(Healthy=false) =P(Happy & NOT Healthy) + P(NOT Happy & NOT Healthy) =0.01 + 0.54 = 0.55

27. CS 573 : Advanced AIProbabilities & Monte-Hall Problem27 Bayes rule P(A|B) = P(A) * P(B|A) / P(B)

28. CS 573 : Advanced AIProbabilities & Monte-Hall Problem28 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(A|B) = P(A & B) / P(B) -> conditional probabilities

29. CS 573 : Advanced AIProbabilities & Monte-Hall Problem29 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(A|B) = P(A & B) / P(B) -> conditional probabilities P(A|B) * P(B) = P(A & B) -> *P(B)

30. CS 573 : Advanced AIProbabilities & Monte-Hall Problem30 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(B|A) = P(A & B) / P(A) -> conditional probabilities P(B|A) * P(A) = P(A & B) -> * P(A) P(A|B) = P(A & B) / P(B) -> conditional probabilities P(A|B) * P(B) = P(A & B) -> *P(B)

31. CS 573 : Advanced AIProbabilities & Monte-Hall Problem31 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(B|A) = P(A & B) / P(A) -> conditional probabilities P(B|A) * P(A) = P(A & B) -> * P(A) P(A|B) = P(A & B) / P(B) -> conditional probabilities P(A|B) * P(B) = P(A & B) -> *P(B)

32. CS 573 : Advanced AIProbabilities & Monte-Hall Problem32 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(B|A) = P(A & B) / P(A) -> conditional probabilities P(B|A) * P(A) = P(A & B) -> * P(A) P(A|B) * P(B) = P(B|A) * P(A) P(A|B) = P(A & B) / P(B) -> conditional probabilities P(A|B) * P(B) = P(A & B) -> *P(B)

33. CS 573 : Advanced AIProbabilities & Monte-Hall Problem33 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(disease|Symptom) Disease Open your abdomen -> painful but accurate based on symptom -> less pain but less accurate

34. CS 573 : Advanced AIProbabilities & Monte-Hall Problem34 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(disease|Symptom) Disease Open your abdomen -> painful but accurate based on symptom -> less pain but less accurate = P(disease) * P(symptom|disease) / P(symptom)

35. CS 573 : Advanced AIProbabilities & Monte-Hall Problem35 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(disease|Symptom) Disease Open your abdomen -> painful but accurate based on symptom -> less pain but less accurate = P(disease) * P(symptom|disease) / P(symptom) Population sampling

36. CS 573 : Advanced AIProbabilities & Monte-Hall Problem36 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(disease|Symptom) Disease Open your abdomen -> painful but accurate based on symptom -> less pain but less accurate = P(disease) * P(symptom|disease) / P(symptom) Population sampling

37. CS 573 : Advanced AIProbabilities & Monte-Hall Problem37 Bayes rule P(A|B) = P(A) * P(B|A) / P(B) P(disease|Symptom) Disease Open your abdomen -> painful but accurate based on symptom -> less pain but less accurate = P(disease) * P(symptom|disease) / P(symptom) Patients sampling

38. CS 573 : Advanced AIProbabilities & Monte-Hall Problem38 The “ Monte-Hall ” Problem

39. CS 573 : Advanced AIProbabilities & Monte-Hall Problem39 The “ Monte-Hall ” Problem Your selection: DOOR 2

40. CS 573 : Advanced AIProbabilities & Monte-Hall Problem40 The “ Monte-Hall ” Problem Your selection: DOOR 2 P(You-get-prize) = 1/3

41. CS 573 : Advanced AIProbabilities & Monte-Hall Problem41 The “ Monte-Hall ” Problem Your selection: DOOR 2 P(You-get-prize) = 1/3 Host open DOOR 1

42. CS 573 : Advanced AIProbabilities & Monte-Hall Problem42 The “ Monte-Hall ” Problem Your selection: DOOR 2 P(You-get-prize) = 1/3 Host open DOOR 1

43. CS 573 : Advanced AIProbabilities & Monte-Hall Problem43 The “ Monte-Hall ” Problem Before DOOR 1 open: P(prize=1) = P(prize=2) = P(prize=3) = 1/3 After DOOR 1 open: P(prize=1)=0, P(prize=2) = P(prize=3) = 1/2

44. CS 573 : Advanced AIProbabilities & Monte-Hall Problem44 The “ Monte-Hall ” Problem There are three objects 1.Empty-A 2.Empty-B 3.Prize Three Doors 1.DOOR-1 2.DOOR-2 3.DOOR-3 1.You choose Empty-A 2.You choose Empty-B 3.You choose Prize

45. CS 573 : Advanced AIProbabilities & Monte-Hall Problem45 The “ Monte-Hall ” Problem There are three objects 1.Empty-A 2.Empty-B 3.Prize Three Doors 1.DOOR-1 2.DOOR-2 3.DOOR-3 1.You choose Empty-A host reveal Empty-B switch->win 2.You choose Empty-B 3.You choose Prize

46. CS 573 : Advanced AIProbabilities & Monte-Hall Problem46 The “ Monte-Hall ” Problem There are three objects 1.Empty-A 2.Empty-B 3.Prize Three Doors 1.DOOR-1 2.DOOR-2 3.DOOR-3 1.You choose Empty-A host reveal Empty-B switch->win 2.You choose Empty-B host reveal Empty-A switch->win 3.You choose Prize

47. CS 573 : Advanced AIProbabilities & Monte-Hall Problem47 The “ Monte-Hall ” Problem There are three objects 1.Empty-A 2.Empty-B 3.Prize Three Doors 1.DOOR-1 2.DOOR-2 3.DOOR-3 1.You choose Empty-A host reveal Empty-B switch->win 2.You choose Empty-B host reveal Empty-A switch->win 3.You choose Prize host reveal Empty-A or B switch->lose

48. CS 573 : Advanced AIProbabilities & Monte-Hall Problem48 The “ Monte-Hall ” Problem Bayes rule explanation

49. CS 573 : Advanced AIProbabilities & Monte-Hall Problem49 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3

50. CS 573 : Advanced AIProbabilities & Monte-Hall Problem50 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2

51. CS 573 : Advanced AIProbabilities & Monte-Hall Problem51 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2 X1 -> O2 X2 -> O1 X3 -> O1 or O2

52. CS 573 : Advanced AIProbabilities & Monte-Hall Problem52 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2 Host open DOOR 1, P(X2 | O1) = ?

53. CS 573 : Advanced AIProbabilities & Monte-Hall Problem53 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2 Host open DOOR 1, P(X2 | O1) = ? P(X2|O1) = P(O1|X2)*P(X2)/P(O1)

54. CS 573 : Advanced AIProbabilities & Monte-Hall Problem54 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2 Host open DOOR 1, P(X2 | O1) = ? P(X2|O1) = P(O1|X2)*P(X2)/P(O1) = 1 *

55. CS 573 : Advanced AIProbabilities & Monte-Hall Problem55 The “ Monte-Hall ” Problem Notation: Oi : Open DOOR i Xi : Prize is behind DOOR i Choose Door 3 P(X2) = 1/3 P(O1)=P(O2)=1/2 Host open DOOR 1, P(X2 | O1) = ? P(X2|O1) = P(O1|X2)*P(X2)/P(O1) = 1 * (1/3) / (1/2) = 2/3