ESE 326 assignment 3 FL2021 PDF

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ESE 326 homework assignment #3 (due 09/27/2021 Monday before class) Part I: Problems from the textbook, chapter 2, starting Page 40 Problems: 17, 19, 27, 32, 35 17. In studying the causes of power failures, these data have been gathered: 5% are due to transformer damage 80% are due to line damage 1% involve both problems Based on these percentages, approximate the probability that a given power failure involves (a) (b) (c) (d) (e)

Line damage given that there is transformer damage Transformer damage given that there is line damage Transformer damage but not line damage Transformer damage given that there is no line damage Transformer damage or line damage

19. Let 𝐴1 𝑎𝑛𝑑 𝐴2 be events such that 𝑃[𝐴1 ] = 0.6, 𝑃[𝐴2 ] = 0.4, 𝑎𝑛𝑑 𝑃[𝐴1 ∪ 𝐴2 ] = 0.8. Are 𝐴1 𝑎𝑛𝑑 𝐴2 independent?

27. The probability that a unit of blood was donated by a paid donor is 0.67. If the donor was paid, the probability of contracting serum hepatitis from the unit is 0.0144, If the donor was not paid, this probability is 0.0012. A patient receives a unit of blood, what is the probability of the patient’s contracting serum hepatitis from this source?

32. Let 𝐴1 𝑎𝑛𝑑 𝐴2 be mutually exclusive events such that 𝑃[𝐴1 ]𝑃[𝐴2 ] > 0. Show that these events are not independent.

35. A test has been developed to detect a particular type of arthritis in individuals over 50 years old. From a national survey it is known that approximately 10% of the individuals in this age group suffer from this form of arthritis. The proposed test was given to individuals with confirmed arthritics disease, and a positive test result was obtained in 85% of the cases. When the test was administered to individuals of the same age group who were known to be free of the disease, 4% were reported to have the disease. What is the probability that an individual has this disease given that the test indicates its presence?

Part II: Problems out of the textbook: Q1: Independence of complements: Show that if two events 𝐴 𝑎𝑛𝑑 𝐵 are independent, then the events 𝐴 𝑎𝑛𝑑 𝐵𝑐 are also independent, where 𝐵𝑐 is the complement of 𝐵.

Q2: Mutually exclusive events and independent events: Show that two mutually exclusive events 𝐴 𝑎𝑛𝑑 𝐵 with 𝑃[𝐴] > 0 𝑎𝑛𝑑 𝑃[𝐵] > 0 are not independent. Q3: Identifying the source of defective item: Three different machines M1, M2, and M3 were used for producing a large batch of similar manufactured items. Suppose that 20 percent of the items were produced by machine M1, 30 percent by machine M2, and 50 percent by machineM3. Suppose further that 1 percent of the items produced by machine M1 are defective, that 2 percent of the items produced by machine M2 are defective, and that 3 percent of the items produced by machineM3 are defective. Finally, suppose that one item is selected at random from the entire batch and it is found to be defective. We shall determine the probability that this item was produced by machine M2....