homework3-winter11-key-1

homework3-winter11-key-1 - Homework 3. • • • • •...

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Unformatted text preview: Homework 3. • • • • • 1. ­ Reliability theory is concerned with the probability that a system will function. When such a system consists of separate components, which may or may not function independently, then the reliability of the system depends in various ways on the reliability of these components. We refer to the reliability of a component as the probability that the component will function for a given period of time. The reliability of the system depends on the reliability of its component parts and the manner in which they are connected. The components could be connected in series, parallel or both. Answer the following questions: (a) A simple computer consists of a processor, a bus, and a memory. The computer will work only if all the three are functioning correctly. The probability that the processor is functioning is 0.99, that the bus in functioning is 0.95, and that the memory is functioning is 0.99. This can be viewed as a series interconnection among three components. What is the probability that the computer will work? Rel= 0.99 (0.95)(0.99) =0.931095 (b) A system consists of five components in series, each having a reliability of 0.97. What is the reliability of the system? Rel=0.975 =0.86 (c) What happens to a series system as the number of components increases? Explain. Reliability quickly decreases as the number of components increases. If we double the number of components from 5 to 10, we would get Rel= 0.9710 =0.737424 2. A simplified model for the movement of the prize of a stock supposes that on each day the stock’s price either moves up 1 unit with probability p or it moves down 1 unit with probability 1 ­p. The changes on different days are assumed independent. Homework must be answered in the order shown here (else please make a note telling the reader where it is). Write your answers neatly Work must be shown for full credit No late homework accepted under any circumstances Homework must be stapled (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock’s price will have increased by 1 unit? (c) Given that after 3 days the stock’s price has increased by 1 unit, what is the probability that it went up on the first day? 4. There is a 50 ­50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50 ­50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier ? If there is a fourth prince, what is the probability that he will have hemophilia? 5. ­ Suppose that each child born to a couple is equally likely to be a boy or a girl independent of the sex distribution of the other children in the family. For a couple having 5 children, compute the probabilities of the following events: (a) All children are of the same sex (b) The three eldest are boys and the other girls (c ) Exactly 3 are boys (d ) The two oldest are girls (e) There is at least 1 girl 6. ­ A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities, given that she is accepted and that she is rejected, of receiving notification on each day of next week are as follows: Day Monday Tuesday Wednesday Thursday Friday She estimates that her probability of being accepted is .6. (a) What is the probability that mail is received on Monday? (b) What is the conditional probability that mail is received on Tuesday given that it is not received on Monday? (c) If there is no mail through Wednesday, what is the conditional probability that she will be accepted? (d) What is the conditional probability that she will be accepted if mail comes on Thursday? (e) What is the conditional probability that she will be accepted if no mail arrives that week? M=mail, A=accept, NA=reject P(mail | accepted) P(mail | rejected) .15 .20 .25 .15 .10 .05 .10 .10 .15 .20 (a) P(M)=P(M|A)P(A)+P(M|NA)P(NA)=0.15*0.6+0.05*0.4=0.11 (b) (c) (d) (e) 7. ­A bin contains 25 lightbulbs, 5 of which are in good condition and will function for at least thirty days, 10 of which are partially defective and will fail in their second day of use, and 10 of which are totally defective and will not light up. Given that a randomly chosen bulb initially lights, what is the probability it will still be working after one week? 8. ­One probability class of 30 students contains 15 that are good, 10 that are fair, and 5 that are of poor quality. A second probability class, also of 30 students, contains 5 that are good, 10 that are fair, and 15 that are poor. You (the expert) are aware of these numbers, but you have no idea which class is which. If you examine one student selected at random from each class and find that the student from class A is a fair student whereas the student from class B is a poor student, what is the probability that class A is the superior class? 9. ­An urn contains 8 red balls and 4 white balls. We draw 2 balls from the urn without replacement. If we assume that at each draw each ball in the urn is equally likely to be chosen, what is the probability that both balls drawn are red? Use product rule to answer. Let R1 denote the event that the first ball is red and let R2 be the event that the second ball is red. Given that the first ball selected is red, there are 7 remaining red balls and 4 white balls. So P(R2|R1)= 7/11 P(R1 R2)= P(R1) P(R2|R1) = (8/12) (7/11)= 14/33 This could have also been computed as ...
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This document was uploaded on 03/20/2011.

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