## Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n

## MA 3351 – Fall 2015 Homework #3 Due Friday 18 September 1. Find eigenvalues and eigenvectors of the following matrices 1 2 2 4 3 1 1 2 3 0 0 4 1 2 1 3 0 −1 1 0 −2 1 0 1 −2 0 0 0 1 0 1 0 −1 0 0 0 0 1 . Do calculations by hand, though you can use Mathematica to check your results. 2. Find eigenvectors and eigenvalues of A = 2 0 1 1 2 −1 0 0 3 . Show that one of the eigenvalues is defective. Do calculations by hand, though you can use Mathematica to check your results. 3. Solve the initial value problem y′ = Ay, y (0) = y0 for the following cases (a) A = −4 1 1 −4 y0 = 1 2 (b) A = −1 1 0 −2 y0 = −1 3 (c) A = 1 0 0 0 −2 1 0 1 −2 y0 = 1 0 2 Do all calculations by hand. 4. Repeat problem 3 using Mathematica to do all calculations. MORE PROBLEMS ON BACK OF PAGE 1 5. Use Mathematica’s Eigensystem function to find eigenvalues and eigenvectors of A = −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 . Suppose you are interested in solutions to y′ = Ay. Without constructing the full solution, answer the following questions: (a) Does the solution grow or decay in time (or a mix of both)? (b) What is the smallest (in magnitude) rate constant? (c) What is the largest (in magnitude) rate constant? (d) As t → ¥, the solution will be dominated by one eigenvector times an exponen- tial. Which eigenvector, and what is the rate constant of the exponential? 6. Use diagonalization to compute (Is − A)−1, where A = −2 1 0 1 −2 1 0 1 −2 . You may use Mathematica. I suggest running FullSimplify on your result. 2

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## A box contains 10 items, of which 3 are defective and 7 are non-defective. Two items are randomly selected, one at a time, with replacement, and x is the number of defectives in the sample of two. Explain why x is a binomial random variable.

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## Mary works in a factory that produces 1,000 telephones each day. When 30 telephones were sampled, it was found that 9 were defective. Estimate how many telephones are defective each day.

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## 1-Two notions serve as the basis for all torts: wrongs and compensation. True False 2-The goal of tort law is to put a defendant in the position that he or she would have been in had the tort occurred to the defendant. True False 3-Hayley is injured in an accident precipitated by Isolde. Hayley files a tort action against Isolde, seeking to recover for the damage suffered. Damages that are intended to compensate or reimburse a plaintiff for actual losses are: compensatory damages. reimbursement damages. actual damages. punitive damages. 4-Ladd throws a rock intending to hit Minh but misses and hits Nasir instead. On the basis of the tort of battery, Nasir can sue: Ladd. Minh. the rightful owner of the rock. no one. 4-Luella trespasses on Merchandise Mart’s property. Through the use of reasonable force, Merchandise Mart’s security guard detains Luella until the police arrive. Merchandise Mart is liable for: assault. battery. false imprisonment. none of the choice 6-The extreme risk of an activity is a defense against imposing strict liability. True False 7-Misrepresentation in an ad is enough to show an intent to induce the reliance of anyone who may use the product. True False 8-Luke is playing a video game on a defective disk that melts in his game player, starting a fire that injures his hands. Luke files a suit against Mystic Maze, Inc., the game’s maker under the doctrine of strict liability. A significant application of this doctrine is in the area of: cyber torts. intentional torts. product liability. unintentional torts 9-More than two hundred years ago, the Declaration of Independence recognized the importance of protecting creative works. True False 10-n 2014, Cloud Computing Corporation registers its trademark as provided by federal law. After the first renewal, this registration: is renewable every ten years. is renewable every twenty years. runs for life of the corporation plus seventy years. runs forever. 11-Wendy works as a weather announcer for a TV station under the character name Weather Wendy. Wendy can register her character’s name as: a certification mark. a trade name. a service mark. none of the choices 12-Much of the material on the Internet, including software and database information, is not copyrighted. True False 13-In a criminal case, the state must prove its case by a preponderance of the evidence. True False 14-Under the Fourth Amendmentt, general searches through a person’s belongings are permissible. True False 15-Maura enters a gas station and points a gun at the clerk Nate. She then forces Nate to open the cash register and give her all the money. Maura can be charged with: burglary. robbery. larceny. receiving stolen property. 16-Reno, driving while intoxicated, causes a car accident that results in the death of Santo. Reno is arrested and charged with a felony. A felony is a crime punishable by death or imprisonment for: any period of time. more than one year. more than six months. more than ten days. 17-Corporate officers and directors may be held criminally liable for the actions of employees under their supervision. True False 18-Sal assures Tom that she will deliver a truckload of hay to his cattle ranch. A person’s declaration to do a certain act is part of the definition of: an expectation. a moral obligation. a prediction. a promise. 19-Lark promises to buy Mac’s used textbook for $60. Lark is: an offeror. an offeree a promisee. a promisor. 20-Casey offers to sell a certain used forklift to DIY Lumber Outlet, but Casey dies before DIY accepts. Most likely, Casey’s death: did not affect the offer. shortened the time of the offer but did not terminated it. extended the time of the offer. terminated the offer.

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## A shipment of 6 refrigerators to a restaurant contains 2 defective ones. The restaurant manager begins to randomly test the 6 refrigerators one at a time. a)Find the probability that the last defective refrigerator is found on the fourth test b)Find the probability that no more than four refrigerators need to be tested to find both of the defective refrigerators.

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## Q1: A small town has two banks A and B. It is estimated that 45% of the potential customers do business only with bank A, 30% only with bank B, and 15% with both banks A and B. The remaining 10% of the customers do business with none of the banks. If E1(E2) denotes the event of a randomly selected customer doing business with bank A(B), find the following probabilities: P(E1), P(E2), P(E1∩E2),P(Ē1Ē2) and P(Ē1UE2) Q2: The inspection of a batch of laminated composite beams produced in a company for defects yielded the following data: No. of defects Proportion of Beams with defects inside Proportion of Beams with defects on surface Total 0 0.4 0.15 0.55 1 0.1 0.05 0.15 2 0.07 0.03 0.1 3 0.06 0.02 0.08 4 0.02 0.03 0.05 5 or more 0.03 0.04 0.07 Total 0.68 0.32 1.0 Determine the probability that the beam has a defect on the surface or it has 4 or more defects. Q3. A batch of 1000 piston rings manufactured in an engine manufacturing facility contains 40% defective. Two piston rings are randomly selected from the batch, one at a time, without replacement. If Ei denotes the event that the i th piston ring selected is defective (i=1, 2), determine the values, P(E1) and P(E2). Q4. An automobile transmission can fail due to three types of problems i.e. gear failure, bearing failure, or shaft failure, wit probabilities 0.3, 0.5 an 0.2 respectively. The probability of transmission failure given a gear failure is 0.5, given a bearing failure is 0.5 and given a shaft failure is 0.6. If a transmission fails, what is the most likely cause? Q5. In the manufacture of a fiber-reinforced laminated composite material, the following probabilities can be associated with the failure of the components made out of this material: Prob. Of failure of components Level of defect in material 0.2 High 0.05 Medium 0.01 Low In a batch of composite material manufactured, 10% of material is found to have High defects, 30% to Medium level defects and 60% to Low level of defects. For a component using this batch of material, indicate the various events associated with the failure of component as a Tree diagram. Also, determine the probability that the component fails.

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## MA 3351 – Fall 2015 Homework #4 Due Friday, 25 September 1. Use the method of variation of parameters to solve y′ = Ay + f (t) y (0) = y0 where (a) A = −2 1 1 −2 f (t) = t 1 y0 = 0 0 (b) A = 4 1 1 4 f (t) = e−t t2 y0 = 1 −1 (c) A = −4 2 1 −5 f (t) = e−t t2 y0 = 1 0 Do all calculations by hand. Use diagonalization to compute any matrix exponentials. Be sure to take advantage of the properties of symmetric matrices when applicable. 2. Repeat problem 1 using Mathematica. Be sure to use MatrixExp instead of Exp when computing matrix exponentials. 3. Use the method of Laplace transforms to solve problem 1, using Mathematica to do the calculations. 4. Let A be the matrix A = 2 1 −1 4 . (a) A has a single eigenvalue l of multiplicity 2. Find it. (b) Find the eigenspace for l. Is A defective? (c) Let J be the Jordan block l 1 0 l . By hand, find a matrix V such that AV = VJ. ADDITIONAL PROBLEMS ON BACK PAGE 1 5. Let J be the Jordan block J = 5 1 0 5 . The solution to y′ = Jy, y (0) = y0 is y (t) = eJty0; however, because J is defective we can’t use diagonalization to compute eJt. Instead, we can compute it using Laplace transforms. (a) By hand, find (Is − J)−1. (b) Find the inverse Laplace transform of (Is − J)−1. You may use Mathematica for this step. (c) Compare your result to eJt as computed by Mathematica’s MatrixExp function. 6. Consider the system y′ = Ay + f (t) with initial conditions y (0) = 0 0 1 0 0 where f (t) = sin (pt) 0 0 0 0 and A = −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 0 0 1 −2 . (a) Use either Laplace transforms or variation of parameters (whichever you prefer) to solve the problem. You should use Mathematica to do your calculations; run FullSimplify on the results. (b) Plot y1 (t) , y2 (t), …, y5 (t) versus t over the interval t ∈ [0, 10]. Show all five functions on the same figure. (To see how to plot multiple functions on the same figure, see theMathematica examples I provided for you as well as theMathematica documentation.) 2

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