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Probability Part 1

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L. Wang, Department of Statistics University of South Carolina; Slide 2 A Few Terms Probability represents a (standardized) measure of chance, and quantifies uncertainty. Probability represents a (standardized) measure of chance, and quantifies uncertainty. Let S = sample space which is the set of all possible outcomes. Let S = sample space which is the set of all possible outcomes. An event is a set of possible outcomes that is of interest. An event is a set of possible outcomes that is of interest. If A is an event, then P(A) is the probability that event A occurs. If A is an event, then P(A) is the probability that event A occurs.

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L. Wang, Department of Statistics University of South Carolina; Slide 3 Identify the Sample Space What is the chance that it will rain today? What is the chance that it will rain today? The number of maintenance calls for an The number of maintenance calls for an old photocopier is twice that for the new old photocopier is twice that for the new photocopier. What is the chance that the photocopier. What is the chance that the next call will be regarding an old next call will be regarding an old photocopier? photocopier? If I pull a card out of a pack of 52 cards, what is the chance its a spade? If I pull a card out of a pack of 52 cards, what is the chance its a spade?

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L. Wang, Department of Statistics University of South Carolina; Slide 4 Union and Intersection of Events The intersection of events A and B refers to the probability that both event A and event B occur. The intersection of events A and B refers to the probability that both event A and event B occur. The union of events A and B refers to the probability that event A occurs or event B occurs or both events, A & B, occur. The union of events A and B refers to the probability that event A occurs or event B occurs or both events, A & B, occur.

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L. Wang, Department of Statistics University of South Carolina; Slide 5 Mutually Exclusive Events Mutually exclusive events can not occur at the same time. Mutually exclusive events can not occur at the same time. Mutually Exclusive EventsNot Mutually Exclusive Events SS

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L. Wang, Department of Statistics University of South Carolina; Slide 6 A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 200 lamps. A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 200 lamps.

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L. Wang, Department of Statistics University of South Carolina; Slide 7 Probability of the Union of Two Events What is the probability that a randomly chosen light will have performed Good in Useful Life? What is the probability that a randomly chosen light will have performed Good in Useful Life? Good in Intensity? Good in Intensity? Good in Useful Life or Good in Intensity? Good in Useful Life or Good in Intensity? Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 8 The Union of Two Events If events A & B intersect, you have to subtract out the double count. If events A & B intersect, you have to subtract out the double count. If events A & B do not intersect (are mutually exclusive), there is no double count. If events A & B do not intersect (are mutually exclusive), there is no double count.

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L. Wang, Department of Statistics University of South Carolina; Slide 9 What is the probability that a randomly chosen light will have performed Good in Intensity or Satisfactorily in Useful life? What is the probability that a randomly chosen light will have performed Good in Intensity or Satisfactorily in Useful life? 130/20 130/20 A. 43/200 B. 173/200 C. 148/200 Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 10 What is the probability that a randomly chosen light will have performed Unsatisfactorily in both useful life and intensity? What is the probability that a randomly chosen light will have performed Unsatisfactorily in both useful life and intensity? A. 2/20 B. 32/200 C. 2/200 D. 4/200 Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 11 Conditional Probability What is the probability that a randomly chosen light performed Good in Useful Life? What is the probability that a randomly chosen light performed Good in Useful Life? Good in Intensity. Good in Intensity. Given that a light had performed Good in Useful Life, what is the probability that it performed Good in Intensity? Given that a light had performed Good in Useful Life, what is the probability that it performed Good in Intensity? Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 12 Conditional Probability Given that a light had performed Good in Intensity, what is the probability that it will perform Good in Useful Life? Given that a light had performed Good in Intensity, what is the probability that it will perform Good in Useful Life? A. 100/145 B. 100/130 C. 100/200 Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 13 Given that a light had performed Good in Intensity, what is the probability that it performed Unsatisfactorily in Useful life? Given that a light had performed Good in Intensity, what is the probability that it performed Unsatisfactorily in Useful life? A. 5/12 B. 5/130 C. 5/200 D. 10/145 Useful Life IntenGoodSatUnsatTotal Good100255130 Sat3510550 Unsat108220 Total1454312200

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L. Wang, Department of Statistics University of South Carolina; Slide 14 Conditional Probability The conditional probability of B, given that A has occurred: The conditional probability of B, given that A has occurred:

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L. Wang, Department of Statistics University of South Carolina; Slide 15 Probability of Intersection Solving the conditional probability formula for the probability of the intersection of A and B: Solving the conditional probability formula for the probability of the intersection of A and B:

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L. Wang, Department of Statistics University of South Carolina; Slide 16 We purchase 30% of our parts from Vendor A. Vendor As defective rate is 5%. What is the probability that a randomly chosen part is defective and from Vendor A? A. 0.200 B. 0.050 C. 0.015 D. 0.030

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L. Wang, Department of Statistics University of South Carolina; Slide 17 We are manufacturing widgets. 50% are red, 30% are white and 20% are blue. What is the probability that a randomly chosen widget will not be white? A. 0.70 B. 0.50 C. 0.20 D. 0.65

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L. Wang, Department of Statistics University of South Carolina; Slide 18 When a computer goes down, there is a 75% chance that it is due to an overload and a 15% chance that it is due to a software problem. There is an 85% chance that it is due to an overload or a software problem. What is the probability that both of these problems are at fault? A. 0.11 B. 0.90 C. 0.05 D. 0.20

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L. Wang, Department of Statistics University of South Carolina; Slide 19 It has been found that 80% of all accidents at foundries involve human error and 40% involve equipment malfunction. 35% involve both problems. If an accident involves an equipment malfunction, what is the probability that there was also human error? A. 0.3200 B. 0.4375 C. 0.8500 D. 0.8750

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L. Wang, Department of Statistics University of South Carolina; Slide 20 Suppose there is no Conditional Relationship between Useful Life & Intensity. What is the probability a light performed Good in Intensity? What is the probability a light performed Good in Intensity? Given that a light had performed Good in Useful Life, what is the probability that it will perform Good in Intensity? Given that a light had performed Good in Useful Life, what is the probability that it will perform Good in Intensity? Useful Life IntenGoodSatUnsatTotal Good1281616160 Sat162220 Unsat162220 Total1602020200

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L. Wang, Department of Statistics University of South Carolina; Slide 21 When, We Say that Events B and A are Independent. The basic idea underlying independence is that information about event A provides no new information about event B. So given event A has occurred, doesnt change our knowledge about the probability of event B occurring.

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L. Wang, Department of Statistics University of South Carolina; Slide 22 There are 10 light bulbs in a bag, 2 are burned out. There are 10 light bulbs in a bag, 2 are burned out. If we randomly choose one and test it, what is the probability that it is burned out? If we randomly choose one and test it, what is the probability that it is burned out? If we set that bulb aside and randomly choose a second bulb, what is the probability that the second bulb is burned out? If we set that bulb aside and randomly choose a second bulb, what is the probability that the second bulb is burned out?

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L. Wang, Department of Statistics University of South Carolina; Slide 23 Near Independence EX: Car company ABC manufactured 2,000,000 cars in 2008; 1,500,000 of the cars had anti-lock brakes. EX: Car company ABC manufactured 2,000,000 cars in 2008; 1,500,000 of the cars had anti-lock brakes. –If we randomly choose 1 car, what is the probability that it will have anti-lock brakes? –If we randomly choose another car, not returning the first, what is the probability that it will have anti-lock brakes?

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L. Wang, Department of Statistics University of South Carolina; Slide 24Independence Sampling with replacement makes individual selections independent from one another. Sampling with replacement makes individual selections independent from one another. Sampling without replacement from a very large population makes individual selection almost independent from one another Sampling without replacement from a very large population makes individual selection almost independent from one another

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L. Wang, Department of Statistics University of South Carolina; Slide 25 Probability of Intersection Probability that both events A and B occur: Probability that both events A and B occur: If A and B are independent, then the probability that both occur: If A and B are independent, then the probability that both occur:

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L. Wang, Department of Statistics University of South Carolina; Slide 26 Test for Independence If, then A and B are independent events. If, then A and B are independent events. If A and B are not independent events, they are said to be dependent events. If A and B are not independent events, they are said to be dependent events.

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L. Wang, Department of Statistics University of South Carolina; Slide 27 Four electrical components are connected in series. The reliability (probability the component operates) of each component is 0.90. If the components are independent of one another, what is the probability that the circuit works when the switch is thrown? A. 0.3600 B. 0.6561 C. 0.7290 D. 0.9000 ABCD

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L. Wang, Department of Statistics University of South Carolina; Slide 28 Complementary Events The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of event A is denoted A. The complement of event A is denoted A. Event A is not event A. Event A is not event A. S: A A The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of event A is denoted A. The complement of event A is denoted A. Event A is not event A. Event A is not event A.

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L. Wang, Department of Statistics University of South Carolina; Slide 29 Mutually exclusive events are always complementary. A. True B. False

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L. Wang, Department of Statistics University of South Carolina; Slide 30 An automobile manufacturer gives a 5- year/75,000-mile warranty on its drive train. Historically, 7% of the manufacturers automobiles have required service under this warranty. Consider a random sample of 15 cars. An automobile manufacturer gives a 5- year/75,000-mile warranty on its drive train. Historically, 7% of the manufacturers automobiles have required service under this warranty. Consider a random sample of 15 cars. If we assume the cars are independent of one another, what is the probability that no cars in the sample require service under the warrantee? If we assume the cars are independent of one another, what is the probability that no cars in the sample require service under the warrantee? What is the probability that at least one car in the sample requires service? What is the probability that at least one car in the sample requires service?

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L. Wang, Department of Statistics University of South Carolina; Slide 31 Consider the following electrical circuit: The probability on the components is their reliability (probability that they will operate when the switch is thrown). Components are independent of one another. The probability on the components is their reliability (probability that they will operate when the switch is thrown). Components are independent of one another. What is the probability that the circuit will not operate when the switch is thrown? What is the probability that the circuit will not operate when the switch is thrown? 0.95

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L. Wang, Department of Statistics University of South Carolina; Slide 32 Probability Rules 1) 0 < P(A) < 1 2) Sum of all possible mutually exclusive outcomes is 1. 3) Probability of A or B: 4) Probability of A or B when A, B are mutually exclusive:

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L. Wang, Department of Statistics University of South Carolina; Slide 33 Probability Rules Continued 4) Probability of B given A: 5) Probability of A and B: 6) Probability of A and B when A, B are independent:

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L. Wang, Department of Statistics University of South Carolina; Slide 34 Probability Rules Continued 7) If A and B are compliments: or

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L. Wang, Department of Statistics University of South Carolina; Slide 35 Consider the electrical circuit below. Probabilities on the components are reliabilities and all components are independent. What is the probability that the circuit will work when the switch is thrown? A 0.90 B 0.90 C 0.95

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L. Wang, Department of Statistics University of South Carolina; Slide 36 The number of maintenance calls for an old photocopier is twice that for the new photocopier. A. Maintenance Call for Old Machine. B. Maintenance Call for New Machine. C. Two maintenance calls in a row for old machine. D. Two maintenance calls in a row for new machine Outcomes Old Machine New Machine Probability 0.67 0.33 Which of the following series of events would most cause you to question the validity of the above probability model?

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