ME 366 Midterm 1 – Flashcards
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S is a sample space of 100 possible 3 digit integers
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(2-7) Provide a reasonable description of the sample space for the following random experiment: A scale that displays two decimal places is used to measure material feeds in a chemical plant in tons. There can be more than one acceptable interpretation of this experiment. Describe any assumptions you make
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a) {5, 6, 7,...} b) {6, 7} c) {0, 1, 2, 3, 4} d) S e) 0 (null set) f) {6, 7} g) {0, 1, 2, 3, 4, 5} h) {5, 6, 7...}
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(2-21) A digital scale that provides weights to the nearest gram is used. The sample space for this experiment is S = {0,1,2,3,..}. Let A denote the event that a weight exceeds 4 grams, let B denote the event that a weight is greater than or equal to 6 grams and less than 8 grams, let C denote the event that a weight is less than or equal to 7 grams. Describe the following events: (a) A ∪ B (b) A ∩ B (c) A' (d) A ∪ B ∪ C (e) (A ∪ C)' (f) A ∩ B ∩ C (g) B' ∩ C (h) A ∪ (B ∩ C)
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a) 83/100 b) 47/100 c) 17/100 d) 40/100 e) 90/100 f) 57/100
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(2-26) Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: -----------------------------------shock resistance -------------------------------------high -----low scratch --------------high ------40 ---------7 resistance ----------low -------43 --------10 Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. If a disk is selected at random, determine the following probabilities: Input your answers in the fractional form (do not simplify). (a) P(A) (b) P(B) (c) P(A') (d) P(A ∩ B) (e) P(A ∪ B) (f) P(A' ∪ B)
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a) {x|x≤ 30} b) {x|x ≥ 40} c) {x|30<x0}
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(2-29) The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events A and B as follows: A = {x|x > 30} and B = {x|x < 40}. Describe each of the following events: (a) A' (b) B' (c) A ∩ B (d) A U B
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3^4 = 81
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(2-34) A wireless garage door opener has a code determined by the up, down, or middle setting of 4 switches. How many outcomes are in the sample space of possible codes?
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(2)*(3)*(6) = 36
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(2-36) In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are 2 machine tools, 3 polishing tools, and 6 painting tools, how many different routings (consisting of machining, followed by polishing, and followed by painting) for a part are possible? Note that machine tools, polishing, and painting tools are distinguishable.
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(8!) _____ = 56 (5!3!)
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(2-40) In a sheet metal operation, 5 notches and 3 bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible? Note that each notch is identical and each bend is identical.
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10 cavities, 5 noncomforming parts, 2 chosen a) P(1 noncomforming part) (5 choose 1)*(5 choose 1) ___________________________ = 0.56 (10 choose 2) b) P(at least one noncomforming parts) 1-P(0) = 1- (5 choose 2)/(10 choose 2) = 0.78
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(2-48) Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 10 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 2 parts from among the 10 at random. 5 cavities are affected by a temperature malfunction that results in parts that do not conform to specifications. (a) What is the probability that the inspector finds exactly 1 nonconforming part? (b) What is the probability that the inspector finds at least 1 nonconforming part?
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a) 3/4 b) 1/4 c) 1/4 d) 3/4 e) 1/4
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(2-58) Each of the possible four outcomes of a random experiment is equally likely. The sample space is a, b, c, d. Let A denote the event a, b, d, and B denote the event d. Determine the following: (a) P(A) (b) P(B) (c) P(A') (d) P(A ∪ B) (e) P(A ∩ B)
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a) S = {1, 2, 3, 4, 5, 6, 7, 8, 9} b) 4/9 c) 7/9
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(2-63) An injection-molded part is equally likely to be obtained from any 1 of 9 cavities on a mold. (a) What is the sample space? (b) What is the probability that a part is from cavity 1, 2, 3, or 4? (c) What is the probability that a part is neither from cavity 3 nor 4?
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10^15 possible number combinations P(valid) = 20*(10^6) ____________ = 20*(10^-9) 10^15
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(2-66) A credit card contains 15 digits between 0 and 9. However, only 20 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number?
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(a) 0.5 + 0.2 + 0.1 = 0.8 (b) 0 (c) 0 (d) 0 (e) 1-0.8 = 0.2
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(2-83) If A, B, and C are mutually exclusive events with P(A) = 0.5, P(B) = 0.2, and P(C) = 0.1 determine the following probabilities: (a) P(A ∪ B ∪ C) (b) P(A ∩ B ∩ C) (c) P(A ∩ B) (d) P((A ∪ B) ∩ C) (e) P(A' ∩ B' ∩ C')
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(a) = 40/100 (b) = 82/100 (c) = No
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(2-85) Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: --------------------------shock resistance -------------------------------high--- low scratch -------high------- 40----- 10 resistance --low ---------32 -----18 (a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high? (b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high? (c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive?
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26 letters + 5 Integers = 31 characters (a) P(A) = 5/31 (b) P(B) = 3/31 (c) P(A∩B) = 0.0156 (d) P(A ∪ B) = P(A)+P(B)-P(A∩B) = 0.242
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(2-90) A computer system uses passwords that are 4 characters and each character is one of the 26 letters (a-z) or 5 integers (1-5). Uppercase letters are not used. Let A denote the event that a password begins with a vowel (either a, e, i, o, u) and let B denote the event that a password ends with an odd number (either 1, 3, or 5). Suppose a hacker selects a password at random. Determine the following probabilities: (a) P(A) (b) P(B) (c) P(A ∩ B) (d) P(A ∪ B)
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(a) P(A) = 72/100 (b) P(B) = 50/100 (c) P(A|B) = 40/50 (d) P(A ∩ B) = 40/100 (e) P(B|A) = 40/72
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(2-99) Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized in the problem above. Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Determine the following probabilities: (a) P(A) (b) P(B) (c) P(A|B) (d) P(A ∩ B) (e) P(B|A)
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(a) = 10/100 (b) = 9/99 (c) = (10/100)(9/99) = 0.0091 (d) = 10/100
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(2-107) A lot of 100 semiconductor chips contains 10 that are defective. Two are selected randomly, without replacement, from the lot. (a) What is the probability that the first one selected is defective? (b) What is the probability that the second one selected is defective given that the first one was defective? (c) What is the probability that both are defective? (d) How does the answer to part (b) change if chips selected were replaced prior to the next selection? Find the probability.
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(a) = P(A|B)P(B) .12 (b) = P(A'|B)P(B) = .6(0.3) = 0.18
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(2-121) Suppose that P(A|B) = 0.4 and P(B) = 0.3 . Determine the following: (a) P(A ∩ B) (b) P(A' ∩ B)
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P(D) = 0.8 P(W) = 0.2 P(F|D) = 0.05 P(F|W) = 0.1 P(F) = P(F|D)P(D) + P(F|W)P(W) = 0.06
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(2-123) The probability is 5% that an electrical connector that is kept dry fails during the warranty period of a portable computer. If the connector is ever wet, the probability of a failure during the warranty period is 10%. If 80% of the connectors are kept dry and 20% are wet, what proportion of connectors fail during the warranty period?
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P(C) = 0.8 P(N) = 0.2 P(F|C) = 0.06 P(F|N) = 0.07 P(F) = P(F|C)P(C) + P(F|N)P(N) = 0.062
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(2-124) Suppose 6% of cotton fabric rolls and 7% of nylon rolls contain flaws. Of the rolls used by a manufacturer, 80% are cotton and 20% are nylon. What is the probability that a randomly selected roll used by the manufacturer contains flaws?
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(a) = P(2nd Defective) = (15/100)(14/99) + (85/100)(15/99) = 0.15 (b) = (15/100)(14/99)(13/98) = 0.0028
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(2-130) A lot of 100 semiconductor chips contains 15 that are defective. (a) Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective. (b) Three are selected, at random, without replacement, from the lot. Determine the probability that all are defective.
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(a) = 11/499 (b) = (12/500)(11/499) =*5.29(10^-4) (c) = (488/500)(487/499) = 0.953 (d) = (10/498) = 0.02 (e) = (11/498) = 0.022 (f) = (12/500)(11/499)(10/498) = 1.06(10^-5)
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(2-108) A batch of 500 containers for frozen orange juice contains 12 that are defective. Two are selected, at random, without replacement from the batch. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable? For the following parts, three containers are selected, at random, without replacement, from the batch. (d) What is the probability that the third one selected is defective given that the first and second one selected were defective? (e) What is the probability that the third one selected is defective given that the first one selected was defective and the second one selected was okay? (f) What is the probability that all three are defective?
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Not independent
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(2-142) If P(A|B) = 0.3, P(B) = 0.7, and P(A) = 0.2, are the events A and B independent?
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Not independent
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(2-144) If P(A) = 0.3, P(B) = 0.6, and A and B are mutually exclusive, are they independent?
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(a) no (b) yes
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(2-145) A batch of 100 containers for frozen orange juice contains 5 that are defective. Two are selected, at random, without replacement, from the batch. Let A and B denote the events that the first and second container selected is defective, respectively. (a) Are A and B independent events? (b) If the sampling were done with replacement, would A and B be independent?
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not independent
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(2-146) Disks of polycarbonate plastics from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: -----------------------------------shock resistance -----------------------------------high------low scratch----------high---------50---------16 resistance------low----------22---------12 Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Are events A and B mutually independent?
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P(Fail) = 0.3 (a) 0.03^2 = 9x10^-4 (b) 1-P(no drive fails) = 1- (0.97^2) = 0.0591
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(2-148) Redundant Array of Inexpensive Discs (RAID) is a technology that uses multiple hard drives to increase the speed of data transfer and provide instant data backup. Suppose that the probability of any hard drive failing in a day is 0.03 and the drive failures are independent. (a) A RAID 0 scheme uses two hard drives each containing a mirror image of the other. What is the probability of data loss? Assume that data loss occurs if both drives fail within the same day. (b) A RAID 1 scheme splits the data over two hard drives. What is the probability of data loss? Assume that data loss occurs if at least one drive fails within the same day.
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(a) (1/10)^3 = 10^-3 (b) (1/10)^3 * (10) = 10^-2 (c) 3* (1/10)^2 * (9/10) = 27*10^-3
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(2-155) 10 cavities in an injection-molding tool produce plastic connectors that fall into a common stream. A sample is chosen every several minutes. Assume that all the samples are independent. (a) What is the probability that 3 successive samples were all produces in cavity 1 of the mold. (b) What is the probability that 3 successive samples were all produced in the same cavity of the mold? (c) What is the probability that 2 out of 3 successive samples were produced in cavity 1 of the mold?
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P(B|A) = [P(A|B)*P(B)]/[P(A)] = (0.6)(0.2)/(0.3) = 0.4
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(2-166) Suppose that P(A|B) = 0.6, P(A) = 0.3, and P(B) = 0.2. Determine P(B|A)?
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P(>2|L) = 0.02 P(>2|F) = 0.05 P(L) = 0.94 P(F) = 0.06 P(>2) = (.02)(.94)+(.05)(.06) = .0218 P(F|>2) = [P(>2|F)*P(F)]/[P(>2)] = (.05)(.06)/(.0218) = 0.138
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(2-168) Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 2% of the legitimate users originate calls from two or more metropolitan areas in a single day. However 5% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 6%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent?
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P(G|H) = .95 P(G|M) = 0.5 P(G|P) = 0.05 P(H) = 0.7 P(M) = 0.2 P(P) = 0.1 (a) P(G) = P(G|H)P(H) + P(G|M)P(M) + P(G|P)P(P) = 0.77 (b) P(H|G) = [P(G|H)P(H)]/[P(G)] = 0.864 P(G') = 1-P(G) = 0.23 P(G'|H) = 1-P(G|H) = .05 (c) P(H|G') = [P(G'|H)P(H)]/[P(G')] = 0.152
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(2-171) Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 50% of moderately successful products received good reviews, and 5% of poor products received good reviews. In addition, 70% of products have been highly successful, 20% have been moderately successful, and 10% have been poor products. (a) What is the probability that a product attains a good review? (b) If a new design attains a good review, what is the probability that it will be a highly successful product? (c) If a product does not attain a good review, what is the probability that it will be a highly successful product?
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{0, 1, 2 ... 50}
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(3-1) The random variable is the number of nonconforming solder connections on a printed circuit board with 50 connections. Determine the range of the random variable.
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a) P(x ≤ 1) = 0.4+0.2+0.1+0.1=0.8 b) P(x>-1) = 0.1+0.1+0.2 = 0.4 c) P(-1≤ x ≤ 0) = 0.2+0.1 = 0.3 d) P(x ≤ -1 or x=1) = 0.4+0.2+0.1 = 0.7
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(3-17) Verify the function is a probability mass function and determine the requested probabilities --------------x -2 -1 0 1 2 ------------f (x) 0.4 0.2 0.1 0.1 0.2 (a) P(X ≤ 1) (b) P(X > −1) (c) P(−1 ≤ X ≤ 0) (d) P(X ≤ −1 or X = 1)
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a) 5/25 b) 4/25 c) 21/25 d) 16/25
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(3-19) The following function is a probability mass function. f (x) = (2x + 1)/25 for x = 0,1,2,3,4 Determine the requested probabilities: (a) P(X = 2) (b) P(X ≤ 1) (c) P(2 ≤ X ≤ 4) (d) P(X > 2)
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f(x) = (5 choose x) (0.6)^x (0.4)^(5-x)
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(3-25) In a semiconductor manufacturing process, 5 wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the probability that a wafer passes the test is 0.6 and that wafers are independent. Determine the probability mass function of the number of wafers from a lot that pass the test.
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f(x) = (5 choose x) (0.2)^x (0.8)^(5-x)
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(3-26) The space shuttle flight control system called PASS (Primary Avionics Software Set) uses 5 independent computers working in parallel. At each critical step, the computers "vote" to determine the appropriate step. The probability that a computer will ask for a roll to the left when a roll to the right is appropriate is 0.2. Let X denote the number of computers that vote for a left roll when a right roll is appropriate. What is the probability mass function of X?
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--x--|--$4 million -- | -- $3 million -- | -- $2 million -- | ______________________________________________________ -f(x)| ------ 0.4 ------ | ------- 0.2 ------- | ------- 0.4 ------- |
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(3-28) Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful, with probabilities 0.4, 0.2, and 0.4, respectively. The yearly revenues associated with a very successful, moderately, or unsuccessful product are $4 million, $3 million, and $2 million, respectively. Let the random variable X denote the yearly revenue of the product. Determine the probability mass function of X.
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f(x=0) = (0.2) (0.3) (0.6) = 0.036 f(x=1) = (0.8) (0.3) (0.6) + (0.2) (0.7) (0.6) + (0.2) (0.3) (0.4) = 0.252 f(x=2) = (0.8) (0.7) (0.6) + (0.8) (0.3) (0.4) + (0.2) (0.7) (0.4) = 0.488 f(x=3) = (0.8) (0.7) (0.4) = 0.224
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(3-30) An assembly consists of 3 mechanical components. Suppose that the probabilities that the first, second, and third components meet specifications are 0.8, 0.7, and 0.4. Assume that the components are independent. Let X be the number of components that meet specifications. Determine the probability mass function of X.
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PRETEND THIS IS A PIECE-WISE ------- { 0, x < 1 ------- { 0.25, 1≤ x < 2 F(x) = { 0.5, 2≤ x< 3 ------- { 0.75, 3≤ x < 4 ------- { 1, x ≥ 4
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(3-38) The sample space of a random experiment is {a, b, c, d}, and each outcome is equally likely. A random variable is defined in the table below. outcome a b c d x 1 2 4 6 Determine the cumulative distribution function of X
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PRETEND THESE ARE BOTH PIECE-WISE ------ { 0.6, $20 mill f(x) = { 0.3, $10 mill ------ { 0.1, $30 mill ^^^^^^^^^^^^^^^^^ this is the probability mass function Cumulative distribution function below ------- { 0, x < $10 mill F(x) = { 0.3, $10 mill ≤ x < $20 mill ------- { 0.9, $20mill ≤ x < $30 mill ------- { 1, $30mill ≤ x
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(3-47) A disk drive manufacturer sells storage devices with capacities of 1 terabyte, 500 gigabytes, and 100 gigabytes with probabilities 0.6, 0.3, 0.1. The revenues associated with the sales in that year are estimated to be $20 million, $10 million, and $30 million, respectively. Let X denote the revenue of storage devices that year. Determine the cumulative distribution function
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a) P(x ≤ 4) = 0.15 b) P(10 ≤ x ≤ 20) = 0 c) P(x <-20) = 0
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(3-49) The following function is a cumulative distribution function. F(x) = 0 x < −15 0.15 −15 ≤ x < 5 0.75 5 ≤ x < 25 1 25 ≤ x Determine the requested probabilities. (a) P(X ≤ 4) (b) P(10 ≤ X ≤ 20) (c) P(X < −20)
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Binomial Distribution with n = 5 and p = 0.95 a) mean = np = 4.75 b_ variance = np(1-p) = 0.2375
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(3-64) An optical inspection system is to distinguish among different part types. The probability of a correct classification of any part is 0.95. Suppose that 5 parts are inspected and that the classifications are independent. Let the random variable X denote the number of parts that are correctly classified. Determine (a) the mean and (b) variance of X.
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X = U[2,3,4,5,6] µ = (2+6)/2 = 4 σ^2 = ((6-2+1)^2-1)/12
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1. (3-76) Let the random variable X have a discrete uniform distribution on the integers 2 ≤ x ≤ 6. Determine the mean, µ, and variance, σ^2, of X.
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X = U[0.1, 0.11, 0.13] µ = E(x) = sum of xf(x), with f(x) = 1/3 (0.1 + 0.11 + 0.13)/3 = 0.1133 V(x) = E(x^2) - µ^2 = (0.1^2 + 0.11^2 + 0.13^2)/3 - (0.1133)^2 = 1.63 * 10^-4
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2. (3-78) Thickness measurements of a coating process are made to the nearest hundredth of a millimeter. The thickness measurements are uniformly distributed with values 0.10, 0.11, 0.13. Determine the mean and variance of the coating thickness for this process.
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X = U[1,2,3,4,5] , Y = U[1, 8, 27, 64, 125] *FOR X* µ = (5+1)/2 = 3 V(x) = ((5-1+1)^2-)/12 = 2 σ = sqrt(V(x)) = 1.414 *FOR Y* µ = (1+8+27+64+125)/5 = 45 V(y) = E(y^2) - (E(y))^2 seen below (1+64+729+4096+15625)/5 - 45^2 = 2078 σ = sqrt(V(y)) =
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3. (3-83) Suppose that X has a discrete uniform distribution on the integers 1 through 5 including 1 and 5. Determine the mean, variance, and standard deviation of the random variable Y = X^3 and compare to the corresponding results for X.
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Binomial Distribution with n = 4 and p = 0.2 a) P(x=2) = (4 choose 2) (0.2)^2 (0.8)^2 = 0.1536 b) P(x≤2) = P(x=0) + P(x=1) + P(x=2) = (4 choose 0) (0.2)^0 (0.8)^4 + (4 choose 1) (0.2) (0.8)^3 + 0.1536 = 0.9728 c) P(x≥1) = 1 - P(x=0) = 0.5904 d) P(1≤x≤3) = P(x=1) + P(x=2) + P(x=3) = 0.5888
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4. (3-95) The random variable X has a binomial distribution with n = 4 and p = 0.2. Determine the following probabilities: (a) P(X = 2) (b) P(X ≤ 2) (c) P(X ≥ 1) (d) P(1 ≤ X ≤ 3)
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Binomial Distribution with n = 10 and p = 0.2 P(0) = (10 choose 0) (0.2)^0 (0.8)^10 = 0.107
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5. (3-100) An electronic product contains 10 integrated circuits. The probability that any integrated circuit is defective is 0.2, and the integrated circuits are independent. The product operates only if there are no defective integrated circuits. What is the probability that the product operates?
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Binomial Distribution a) n = 4, p = 0.3 P(1) = (4 choose 1) (0.3) (0.7)^3 = 0.4116 b) n = 6, p = 0.3 P(2) = (6 choose 2) (0.3)^2 (0.7)^4 = 0.3241 c) n = 8, p = 0.3 P(x < 3) = P(0) + P(1) + P(2) = 0.5518
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6. (3-103) A particularly long traffic light on your morning commute is green 30% of the time that you approach it. Assume that each morning represents an independent trial. (a) Over 4 mornings, what is the probability that the light is green on exactly 1 day? (b) Over 6 mornings, what is the probability that the light is green on exactly 2 days? (c) Over 8 mornings, what is the probability that the light is green on less than 3 days?
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Geometric Distribution with µ = 5 = 1/p, so p = 1/5 f(x) = (1-p)^(x-1)(p) a) P(x=2) = (0.8) (0.2) = 0.16 b) P(x≤4) = P(1) + P(2) + P(3) + P(4) 0.2 + 0.8(0.2) + (0.8)^2 (0.2) + (0.8)^3 (0.2) = 0.5904 c) P(x>4) = 1 - P(x ≤ 4) = 1 - 0.5904 = 0.4096
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7. (3-120) Suppose the random variable X has a geometric distribution with a mean of 5. Determine the following probabilities: (a) P(X = 2) (b) P(X ≤ 4) (c) P(X > 4)
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a) Geometric Distribution E(x) = 1/p, therefore E(x) = 2.5 b) Negative Binomial Distribution r = 6-4 = 2, p = 0.4 E(x) = r/p = 5
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8. (3-121) Consider a sequence of independent Bernoulli trials with p = 0.4. (a) What is the expected number of trials to obtain the first success? (b) After the 4th success occurs, what is the expected number of trials to obtain the 6th success?
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a) P(4) = (0.1)^3 (0.9) = 9*10^-4 b) P(x ≤ 6) = P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 0.999999 c) P(x ≥ 3) = 1 - P(1) - P(2) = 1 - 0.9(1 + 0.1) = 0.01
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9. (3-123) The probability of a successful optical alignment in the assembly of an optical data storage product is p = 0.9. Assume the trials are independent. (a) What is the probability that the first successful alignment requires exactly 4 trials? (b) What is the probability that the first successful alignment requires at most 6 trials? (c) What is the probability that the first successful alignment requires at least 3 trials?
