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- 1. Outline Deﬁnitions Approaches Axioms Sample-Point Approach 1. Sets and Probability 1.3 Probabilistic Model of an Experiment 1.4 Sample-Point Approach in Calculating Probability Ruben A. Idoy, Jr. Introduction to Probability Theory (Math 181) 21 June 2012
- 2. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachOutline 1 Deﬁnitions
- 3. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachOutline 1 Deﬁnitions 2 Approaches of Probability Values
- 4. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachOutline 1 Deﬁnitions 2 Approaches of Probability Values 3 Axioms of Probability
- 5. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachOutline 1 Deﬁnitions 2 Approaches of Probability Values 3 Axioms of Probability 4 Sample-Point Approach on Calculating Probability Steps Examples
- 6. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions experiment - the process of making an observation.
- 7. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions experiment - the process of making an observation. An experiment can result in one, and only one, of a set of distinctly diﬀerent observable outcomes.
- 8. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions experiment - the process of making an observation. An experiment can result in one, and only one, of a set of distinctly diﬀerent observable outcomes. We are interested in experiments that generate outcomes which vary in random manner and cannot be predicted with certainty.
- 9. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions experiment - the process of making an observation. sample space - denoted by S (or Ω in some books), is a set of points corresponding to all distinctly diﬀerent possible outcomes of an experiment. Each point corresponds to a particular single outcome.
- 10. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions experiment - the process of making an observation. sample space - denoted by S (or Ω in some books), is a set of points corresponding to all distinctly diﬀerent possible outcomes of an experiment. Each point corresponds to a particular single outcome. sample point - a single point in a sample space, S
- 11. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions sample space - denoted by S (or Ω in some books), is a set of points corresponding to all distinctly diﬀerent possible outcomes of an experiment. Each point corresponds to a particular single outcome. Discrete sample space - one that contains a ﬁnite number or countable inﬁnity of sample points.
- 12. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions sample space - denoted by S (or Ω in some books), is a set of points corresponding to all distinctly diﬀerent possible outcomes of an experiment. Each point corresponds to a particular single outcome. Discrete sample space - one that contains a ﬁnite number or countable inﬁnity of sample points. Continuous sample space - has an inﬁnite number of sample points.
- 13. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points.
- 14. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment
- 15. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}),
- 16. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}), B: observe a number less than 5 (B = {1, 2, 3, 4}),
- 17. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}), B: observe a number less than 5 (B = {1, 2, 3, 4}), C: observe a 2 or a 3 (C = {2, 3}),
- 18. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}), B: observe a number less than 5 (B = {1, 2, 3, 4}), C: observe a 2 or a 3 (C = {2, 3}), E1 : observe a 1 (E1 = {1}),
- 19. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}), B: observe a number less than 5 (B = {1, 2, 3, 4}), C: observe a 2 or a 3 (C = {2, 3}), E1 : observe a 1 (E1 = {1}), E6 : observe a 6 (E6 = {6})
- 20. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. Example: Die-tossing Experiment A: observe an odd number (A = {1, 3, 5}), B: observe a number less than 5 (B = {1, 2, 3, 4}), C: observe a 2 or a 3 (C = {2, 3}), E1 : observe a 1 (E1 = {1}), E6 : observe a 6 (E6 = {6}) Each of these 5 events is a speciﬁc collection of sample points.
- 21. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. A simple event is one that contains a single sample point. We may refer to simple events as events that cannot be decomposed.
- 22. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. A simple event is one that contains a single sample point. We may refer to simple events as events that cannot be decomposed. Probability - a numerical measure of the chance of the occurrence of an event.
- 23. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachDeﬁnitions event - any subset of the sample space, S. It can also be viewed as a collection of sample points. A simple event is one that contains a single sample point. We may refer to simple events as events that cannot be decomposed. Probability - a numerical measure of the chance of the occurrence of an event. The ﬁnal step in constructing a probabilistic model for an experiment with a discrete sample space is to attach a probability to each sample event.
- 24. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachApproaches to the Assignment of Probability Values
- 25. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachApproaches to the Assignment of Probability Values Relative Frequency or A Posteriori Approach The probability value is the relative frequency of the occurrence of the event over a long-run experiment (over a large number of repetitions of the experiment).
- 26. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachApproaches to the Assignment of Probability Values Relative Frequency or A Posteriori Approach The probability value is the relative frequency of the occurrence of the event over a long-run experiment (over a large number of repetitions of the experiment). number of times the event occurred P (E) = number of repetitions of the experiment
- 27. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachApproaches to the Assignment of Probability Values Relative Frequency or A Posteriori Approach The probability value is the relative frequency of the occurrence of the event over a long-run experiment (over a large number of repetitions of the experiment). number of times the event occurred P (E) = number of repetitions of the experiment Classical, Theoretical or A Priori Approach Probability value us based on an experimental model with certain assumptions
- 28. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachApproaches to the Assignment of Probability Values Relative Frequency or A Posteriori Approach The probability value is the relative frequency of the occurrence of the event over a long-run experiment (over a large number of repetitions of the experiment). Classical, Theoretical or A Priori Approach Probability value us based on an experimental model with certain assumptions Subjective Approach The researcher assigns probability according to his knowledge or experience on the occurrence of the event. There is no objective way of prediction of the occurrence of the event under this approach.
- 29. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachAxioms of Probability For every event E in a sample space S, we assign a numerical value P(E), known as the probability of E, such that:
- 30. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachAxioms of Probability For every event E in a sample space S, we assign a numerical value P(E), known as the probability of E, such that: 1 P(E) 0;
- 31. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachAxioms of Probability For every event E in a sample space S, we assign a numerical value P(E), known as the probability of E, such that: 1 P(E) 0; 2 P(S) = 1;
- 32. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachAxioms of Probability For every event E in a sample space S, we assign a numerical value P(E), known as the probability of E, such that: 1 P(E) 0; 2 P(S) = 1; 3 If E1 , E2 , . . . form a sequence of pairwise mutually exclusive events in S (Ei ∩ Ej = ∅, i j), then ∞ P(E1 ∪ E2 ∪ E3 ∪ · · · ) = P(Ai ) i=1
- 33. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Let A be the event of obtaining a number less than or equal to 3 in tossing a die.
- 34. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Let A be the event of obtaining a number less than or equal to 3 in tossing a die. Find the probability of A if:
- 35. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Let A be the event of obtaining a number less than or equal to 3 in tossing a die. Find the probability of A if: 1 the die is fair;
- 36. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Let A be the event of obtaining a number less than or equal to 3 in tossing a die. Find the probability of A if: 1 the die is fair; 2 the die is biased such that an odd number is twice as likely to occur as an even number.
- 37. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Solution for [1] First note that S = {1, 2, 3, 4, 5, 6}. Since the die is fair, the probability for each simple event is equal, say p. That is, P(1) = P(2) = · · · = P(6) = p. We further observe that P(1) + P(2) + · · · + P(6) = 1. Substituting p to each probability of the simple event, we get p + p + p + p + p + p = 6p = 1. 1 Thus, p = 6 .
- 38. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Solution for [1] The event A = {1, 2, 3}, has therefore a probability: 1 1 1 3 P(A) = P(1) + P(2) + P(3) = + + = 6 6 6 6
- 39. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Solution for [2] The sample space of the experiment is still the set S = {1, 2, 3, 4, 5, 6}. Let p be the probability of each even number to occur and 2p be the probability of each odd number to occur. That is, P(2) + P(4) + P(6) =p P(1) + P(3) + P(5) =2p Substituting each probability to the simple event, we get 2p + p + 2p + p + 2p + p = 9p = 1. Thus, p = 1 . 9
- 40. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExample Solution for [2] The event A = {1, 2, 3}, has therefore a probability: 2 1 2 5 P(A) = P(1) + P(2) + P(3) = + + = 9 9 9 9 Not all problems dealing with probability of an event are solvable by simply using the Axioms of Probability. Thus, there are 2 ways or approaches known to calculate the Probability of an Event: the sample-point approach and the event-composition method.
- 41. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps:
- 42. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps: 1 Deﬁne the experiment.
- 43. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps: 1 Deﬁne the experiment. 2 List the simple events associated with the experiment and test each to make certain that they cannot be decomposed. This deﬁnes the sample space, S.
- 44. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps: 1 Deﬁne the experiment. 2 List the simple events associated with the experiment and test each to make certain that they cannot be decomposed. This deﬁnes the sample space, S. 3 Assign reasonable probabilities to the sample points in S, making certain that P(Ei ) = 1 S .
- 45. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps: 1 Deﬁne the experiment. 2 List the simple events associated with the experiment and test each to make certain that they cannot be decomposed. This deﬁnes the sample space, S. 3 Assign reasonable probabilities to the sample points in S, making certain that P(Ei ) = 1 S . 4 Deﬁne the event of interest, E, as a speciﬁc collection of sample points.
- 46. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachStepsSample-Point Approach on Calculating Probability Steps: 1 Deﬁne the experiment. 2 List the simple events associated with the experiment and test each to make certain that they cannot be decomposed. This deﬁnes the sample space, S. 3 Assign reasonable probabilities to the sample points in S, making certain that P(Ei ) = 1 S . 4 Deﬁne the event of interest, E, as a speciﬁc collection of sample points. 5 Find P(E) by summing the probabilities of the sample points in E.
- 47. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Example 1 Toss a coin 3 times and observe the top face. What is the probability of observing exactly 2 heads, assuming the coin is fair?
- 48. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Tossing a fair coin 3 times.
- 49. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Tossing a fair coin 3 times. 2 List of simple events: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
- 50. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Tossing a fair coin 3 times. 2 List of simple events: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 3 Assignment of probability to each sample points: 1 P(Ei ) = , i = 1, 2, . . . , 8. 8
- 51. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Tossing a fair coin 3 times. 2 List of simple events: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 3 Assignment of probability to each sample points: 1 P(Ei ) = , i = 1, 2, . . . , 8. 8 4 Deﬁne event of interest: Let A be the event that 2 heads will appear after tossing the coin 3 times.
- 52. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Tossing a fair coin 3 times. 2 List of simple events: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 3 Assignment of probability to each sample points: 1 P(Ei ) = , i = 1, 2, . . . , 8. 8 4 Deﬁne event of interest: Let A be the event that 2 heads will appear after tossing the coin 3 times. 5 Find P(A): 1 1 1 3 P(A) = P(HHT) + P(HTH) + P(THH) = + + = 8 8 8 8
- 53. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Example 2 Patients arriving at a hospital outpatient clinic can select any of three service counters. Physicians are randomly assigned to the stations and the patients have no station preference. Three patients arrived at the clinic and their selection is observed. Find the probability that each station receives a patient.
- 54. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Assigning patients to service counters.
- 55. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Assigning patients to service counters. 2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is, each patient could be assigned to any of the service counter 1,2 and 3. Furthermore, |S| = 33 = 27.
- 56. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Assigning patients to service counters. 2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is, each patient could be assigned to any of the service counter 1,2 and 3. Furthermore, |S| = 33 = 27. 3 Since each simple events are likely to occur, then 1 1 P(Ei ) = = , ∀i = 1, 2, . . . , 27 |S| 27
- 57. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Assigning patients to service counters. 2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is, each patient could be assigned to any of the service counter 1,2 and 3. Furthermore, |S| = 33 = 27. 3 Since each simple events are likely to occur, then 1 1 P(Ei ) = = , ∀i = 1, 2, . . . , 27 |S| 27 4 Deﬁne event of interest: Let B be the event that each station receives a patient.
- 58. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Solution 1 Experiment: Assigning patients to service counters. 2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is, each patient could be assigned to any of the service counter 1,2 and 3. Furthermore, |S| = 33 = 27. 3 Since each simple events are likely to occur, then 1 1 P(Ei ) = = , ∀i = 1, 2, . . . , 27 |S| 27 4 Deﬁne event of interest: Let B be the event that each station receives a patient. 5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · · + P((3, 2, 1)) 1 1 1 6 = 27 + 27 + · · · + 27 = 27
- 59. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamples Example 3 Four cards are drawn from a standard deck of 52 cards. What is the probability that the cards drawn are: 1 of the same suit; 2 of the same color; 3 of the same type.
- 60. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamplesAssignment 1 Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper.
- 61. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamplesAssignment 1 Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper. A box contains seven laptops. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and then classiﬁed as defective or nondefective.
- 62. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamplesAssignment 1 Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper. A box contains seven laptops. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and then classiﬁed as defective or nondefective. (i) Find the probability of the event A that the selection includes no defective.
- 63. Outline Deﬁnitions Approaches Axioms Sample-Point ApproachExamplesAssignment 1 Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper. A box contains seven laptops. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and then classiﬁed as defective or nondefective. (i) Find the probability of the event A that the selection includes no defective. (ii) Find the probability of the event B that the selection includes exactly one defective.

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