## 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

## PROBLEM 2 (20 points) In the following problem we consider two men trying to slide a 6m long plank over an overhead rack. The plank has a mass of 100kg and the coefficient of kinetic friction between the plank and each support is 0.5.

## The inductor in problem 5 is connected in series to a 0.5 W and a battery whose emf is 24 V. What current flows in this circuit 0.1 seconds after the switch is closed?

## Question 2 (1 point) Which of the following is correct about interpreting the results of statistical tests? Question 2 options: 1) Obtaining a probability value of .05 tells us the difference between groups is definitely not caused by chance fluctuation. 2) If a probability value falls above .05, then the results will have to be replicated before we can have confidence in them. 3) Obtaining a probability value of .05 gives us confidence that the findings are not the result of chance, but does not eliminate this possibility. 4) A .05 probability value means there is a 5 percent chance the finding reflects a real difference. Question 3 (1 point) Which of the following statements is true about theories of personality? Question 3 options: 1) They provide only a part of the picture of human personality. 2) They support the expert’s viewpoint. 3) Theories are predicted from one hypothesis or another. 4) They are directly tested using empirical methods. Question 4 (1 point) Which of the following statements is correct about hypothetical constructs? Question 4 options: 1) They are useful inventions by researchers that have no physical reality. 2) They are easier to measure than personality variables. 3) They cannot be measured with personality tests. 4) They have poor reliability and validity. Question 5 (1 point) According to the “law of parsimony,” Question 5 options: 1) a good theory generates a large number of hypotheses. 2) the best theory is the one that explains a phenomenon with the fewest constructs. 3) hypotheses are generated from theories. 4) theories should require as few studies as possible to support them. ________________________________________ Question 6 (1 point) Which of the following does a correlation coefficient not tell us? Question 6 options: 1) If the difference between two means reflects a real difference or can be attributed tochancefluctuation. 2) The strength of a relationship between two measures. 3) The direction of a relationship between two measures. 4) How well a score on one measure can be predicted by a score on another measure. Question 7 (1 point) A researcher finds that males make fewer errors than females when working in a competitive situation. However, women make fewer errors than men when working in acooperative situation. This is an example of Question 7 options: 1) a confound. 2) two manipulated independent variables. 3) an interaction. 4) a failure to replicate.

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## Design a regulated power supply able to supply a maximum load current of 25 mA. Your design specifications are: DC Output: 12 V Load Variation: 0.5 – 2 kilo Ohms Ripple factor < 0.1% for 1 kilo ohm load Load Regulation: Better than 1mV/mA Line Regulation: Better than 2mV/V In your design use diodes 1N4001 and Zener 1N4742. In your design you must mention the values of the resistors, capacitors, transformer turns ratio. Typed report must contain the following in the same order: • Final design with all component values • SPICE simulation results supporting the validity of your design satisfying the specifications. • Discuss your approach to satisfy the specifications. Explain which specifications you met and which you did not meet. Provide discussion and conclusions and comments.

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## Vermont Technical College Electronic Applications ELT-2060 Lab 05: DC characteristics, input offset voltage and input bias current Reference: Operational Amplifiers with Linear Integrated Circuits Fourth edition William D. Stanley, pages 154-155 (Problems 3-21, 3-22 and Lab exercises LE 3-1 to LE 3-4) For the following exercises, make sure to record all calculations, estimations and measured results. Components: 2 741 Op Amps, 10k Ω Potentiometer, 4-10kΩ, 1kΩ , 100kΩ , 100Ω , 560kΩ , 5.6M Ω, resistors Objectives: a. Voltage offset Null Circuit and Closed-loop Differential Circuit b. Measurement of dc Input Offset Voltage c. Measurement of dc Bias and Offset Currents a. Voltage offset Null Circuit and Closed-loop Differential Circuit In this exercise, investigate the use of a null circuit to reduce the output dc offset to its minimum possible value. Refer to the “Voltage Offset Null Circuit” describe in the 741 op amp data sheet from Appendix C of your text book. Although there are no specific closed-loop configurations shown, use a closed-loop differential Amplifier shown in Figure 1. The differential nature of this type of circuit makes it particularly sensitive, therefore well suited, to illustrate the concept dc voltage offset. 1. Connect the closed-loop difference amplifier of Figure 1 with R=10k Ω and A=1. Using a 10kΩ potentiometer connect the “Voltage Offset Null Circuit” between nodes 1 and 5 as shown in the 741 data sheet. Keep in mind that a potentiometer is a three terminal device. You will need to connect the potentiometer wiper terminal to the lowest potential in the circuit -VCC. 2. Connect the two external circuit inputs (v1 and v2) to ground, measure the dc voltage. From the data sheet the expected value of offset voltage at room temperature is 2mV typical and 6mV maximum. Voltages at these levels will be hard to measure with the laboratory multimeter. 3. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 5. Do not break down you difference amplifier. Next, build the non-inverting amplifier as shown in figure 2 with Ri=1k Ω and Rf =100k Ω. Attach the output of the difference amplifier to the input of the non-inverting amplifier. This will amplify your offset by 101. 6. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 7. In effect we amplified the voltage offset from the difference amplifier by 101. Please describe any possible flaws in using this approach. Compare this result to what was measured in step 2. 8. Write an equation that expresses the expected output voltage Vo in terms of the two input voltages V1 and V2. 9. Apply dc input voltage for the following six combinations, compare the results to the expected values you calculate with the equation from step 8 a. V1=10V, V2=0V b. V1=0V, V2=10V c. V1=V2=10V d. V1=10mV, V2=0 e. V1=0, V2=10mV f. V1=V2=10mV b. Measurement of dc Input Offset Voltage ( Stanley Problem 3-21 page 151) A circuit and equation to measure the input offset voltage Vio is show in figure 3. With the proper selection of resistors Ri, Rf, and Rc the effects of offset due to input bias currents can be neglected. When the input terminals are both held to ground the resulting output voltage should be a direct measurement of Vio. 1. Build the circuit in Figure 3 with Ri=100 Ω and Rf=10k Ω measure and record Vo. Compare your results with the specification of input offset voltage provided in the data sheet. 2. Increase the value of Rf to 100k Ω, and measure Vo again. Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input bias currents are negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on how there relationship demonstrates that neglecting input bias current was a valid assumption. c. Measurement of dc Bias and Offset Currents (Stanley Problem 3-22 page 152) Consider the three circuits of figure 4 .The resistance R is chosen large so that the contribution to the output from bias currents is considerably larger than the contribution from the input offset voltages. The accompanying equations will predict the values of Ib+, Ib- and Iio. 1. Start with setting R=560k Ω and build each circuit in figure 4 one at a time. Going from one configuration to the next configuration should be quick, all that is changing is the placement of the resistors. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet. 2. Increase the value of R to 5.6M Ω. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet and to the results in part 1.Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input offset voltage effect is negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on why neglecting input offset voltage was a valid assumption. LAB write up: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. No MultiSim will be required for this report. Please include a written English language paragraph for all lab steps that required an explanation.

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## Author Name: BIO 218 Natural History Paper General Formatting: (10%) • 1 Margins correct? • 1 Font correct? • 2 Double-spaced? • 2 Pages numbered? • 2 All sections included? • 2 At least 3 pages of text, not more than 5 pages? Project elements (50%) • Introduction: o 8 General background on topic and species (state scientific name!)? o 2 Goes from general to specific? • Review of Journal Articles: o 4 States topic and hypothesis/hypotheses described in articles? o 3 Reports how research was conducted? o 2 Describes specialized materials used? o 2 Discusses type(s) of data collected and how to be analyzed/compared/used? o 3 Reports what happened in the experiments? o 2 If comparisons made, discusses how they were made? o 2 Figure(s) reproduced and cited? o 2 Table(s) reproduced and cited? • Summary/Conclusion: o 10 Synthesizes the results of the experiments and ties the findings of the articles together? • Literature Cited: o 4 At least 3 journal articles (primary literature) used? o 2 References used in paper properly? o 2 References all listed in this section and formatted correctly? o 2 All references listed are in the body of the paper and all references in the body are listed in this section? *0.5% for each extra citation (>3) that is correctly used* Writing Elements (40%) • /15 Grammar or spelling errors? • /15 Writing is clear and flows logically throughout paper? • /10 Appropriate content in each section? Final Paper Total ( %) = /40 Comments:

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## A 50 V battery is connected across a 10 ohm resistor and a current of 4 A flows. The internet resistance of the battery is ; A) Zero, B) 0.5 ohm, C) 1.1 ohm, D) 2.5 ohm, E) 5ohm

## A concave lens forms a virtual image 0.5 times the size of the object. The object distance is 13.2 cm. Find the focal length of the lens.

## Tornado Eddy Investigation Abstract The objective of this lab was to write a bunch of jibberish to provide students with a formatting template. Chemical engineering, bioengineering, and environmental engineering are “process engineering” disciplines. Good abstracts contains real content, such as 560 mL/min, 35 deg, and 67 percent yield. Ideal degreed graduates are technically strong, bring broad system perspectives to problem solving, and have the professional “soft skills” to make immediate contributions in the workplace. The senior lab sequence is the “capstone” opportunity to realize this ideal by integrating technical skills and developing professional soft skills to ensure workforce preparedness. The best conclusions are objective and numerical, such as operating conditions of 45 L/min at 32 deg C with expected costs of $4.55/lb. Background Insect exchange processes are often used in bug filtration, as they are effective at removing either positive or negative insects from water. An insect exchange column is a packed or fluidized bed filled with resin beads. Water flows through the column and most of the insects from the water enter the beads, but some of them pass in between the beads, which makes the exchange of insects non-ideal. Insectac 249 resin is a cation exchange resin, as it is being used to attract cationic Ca2+ from the toxic waste stream. This means the resin is negatively charged, and needs to be regenerated with a solution that produces positively charged insects, in this case, salt water which contains Na+ insects. The resin contains acidic styrene backbones which capture the cationic insects in a reversible process. A curve of Ca2+ concentration concentration vs. time was obtained after a standard curve was made to determine how many drops from the low cost barium test kit from Aquarium Pharmaceuticals (API)1 bottle #2 would correspond to a certain concentration in solution. A standard curve works by preparing solutions with known concentrations and testing these concentrations using the kit to create a curve of number of drops from bottle #2 (obtained result) vs. concentration of Ca2+ in solution (desired response). The standard curve can then be used for every test on the prototype and in the field, to quickly and accurately obtain a concentration from the test kit. The barium concentration vs. time curve can be used to calculate the exchange capacity of the resin and, in later tests, the regeneration efficiency. The curves must be used to get the total amount of barium removed from the water, m. Seen in Equation 2, the volumetric flow rate of water, , is multiplied by the integral from tinitial to tfinal of the total concentration of Ca2+ absorbed by the resin as a function of time, C. (2) 1 http://aquariumpharm.com/Products/Product.aspx?ProductID=72 , date accessed: 11/26/10 CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 9 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A graphical trapezoid method was used to evaluate the integral and get the final solution in equivalents of Ca2+ per L, it must be noted that there are 2 equivalents per mole of barium, as the charge of the barium insect is +2. An initial exchange capacity was calculated for the virgin resin, and an adjusted exchange capacity was calculated once the resin was regenerated. The regenerated resin capacity was found by multiplying the virgin resin capacity by the regeneration efficiency, expressed in Equation 3. (3) See Appendix A for the calculation of the exchange capacities and the regeneration efficiency. Materials and Methods Rosalie and Peter Johnson of Corvallis established the Linus Pauling Chair in Chemical Engineering to honor Oregon State University’s most famous graduate. Peter Johnson, former President and owner of Tekmax, Inc., a company which revolutionized battery manufacturing equipment, is a 1955 graduate of the College of Engineering.2 The Chair, also known as the Linus Pauling Distinguished Engineer or Linus Pauling Engineer (LPE), was originally designed to focus on the traditional “capstone” senior lab sequence in the former Department of Chemical Engineering. The focus is now extended to all the process engineering disciplines. The LPE is charged with establishing strong ties with industry, ensuring current and relevant laboratory experiences, and helping upperclass students develop skills in communication, teamwork, project management, and leadership. Include details about lab procedures not sufficiently detailed in the SOP, problems you had, etc. The bulk solution prepared to create the standard curve was used in the second day of testing to obtain the exchange capacity of the insectac 249 resin. The solution was pumped through a bathroom scale into the prototype insect exchange column. 45 mL of resin was rinsed and added to the column. The bed was fluidized as the solution was pumped through the resin, but for the creation of the Ca2+ concentration vs. time curve, the solution was pumped down through the column, as illustrated in the process flow diagram seen in Figure 1. Figure 1. Process sketch of the insect exchange column used for the project. Ref: http://www.generon.co.uk/acatalog/Chromatography.html 2 Harding, P. Viscosity Measurement SOP, Spring, 2010. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 10 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A bathroom scale calibration curve was created to ensure that the 150 mL/min, used to calculate the breakthrough time, would be delivered to the resin. The bathroom scale used was a Dwyer brand with flowrates between 0 and 300 cc/min of water. Originally, values between 120 and 180 mL/min were chosen for the calibration, with three runs for each flowrate, however the bathroom scale values were so far away from the measure values the range was extended to 100 to 200 mL/min. The regeneration experiment was performed using a method similar to that used in the water softening experiment, however instead of using a 640 ppm Ca2+ solution to fill the resin, a 6000 ppm Na+ solution was used to eject the Ca2+ from the resin. Twelve samples times were chosen and adjusted as the experiment progressed, with more than half of the samples taken at times less than 10 minutes, and the last sample taken at 45 minutes. The bulk exit solution was also tested to determine the regeneration efficiency. Results and Discussion The senior lab sequence has its roots in the former Department of Chemical Engineering. CHE 414 and 415 were taught in Winter and Spring and included 6 hours of lab time per week. The School has endeavored to incorporate the courses into the BIOE and ENVE curriculum, and this will be complete in 2008-2009. Recent development of the senior lab course sequence is shown chronologically in Fig. 1. In 2006-2007, CHE 414 and 415 were moved to Fall and Winter to enable CHE 416, an elective independent senior project course. Also that year, BIOE students took BIOE 414 in the Fall and BIOE 415 was developed and taught. No BIOE students enrolled in the optional CHE. In 2007-2008, the program transitioned in a new Linus Pauling Engineer and ENVE 414 was offered. Also, approximately 30 percent of BIOE students enrolled in the optional CHE 416. Accommodating the academic calendars of the three disciplines required a reduction in weekly student lab time from 6 to 3 hours. The expected relationship between coughing rate, y, and length of canine, x, is Bx z y Fe− (1) where F is a pre-exponential constant, B is vitamin B concentration and z is the height of an average trapeze artist. 3 The 2008-2009 brings the challenge of the dramatic enrollment increase shown in Fig. 1 and the first offering of ENVE 415. The result, shown on the right in Fig. 1, is the delivery of the senior lab sequence uniformly across the process engineering disciplines. CBEE 416 is expected to drawn approximately of the students that take the 415 courses. In 2007-2008, 414 and 415 were required for CHEs, 414 and 415 for BIOEs, and only 414 for ENVEs. CHE 416 is ostensibly an elective for all disciplines. In 2008-2009, 414 and 415 is required for all disciplines and CHE 416 will be an elective. The content of 414 is essentially 3 Fundamentals of Momentum, Heat, and Mass Transfer, Welty, J.R. et al., 4th edition, John Wiley & Sons, Inc. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 11 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE identical for all three disciplines, 415 has discipline-specific labs, and 416 consists of senior projects with potentially cross-discipline teams of 2 to 4 students. Tremendous labor and struggling with the lab equipment resulted in the data shown in y = –‐0.29x + 1.71 y = –‐0.25x + 2.03 y = –‐0.135x + 2.20 –‐1.5 –‐1.0 –‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ln y (units) x (units) ln y_1 ln y_2 ln y_3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Case 1 Case 2 Case 3 Slope (units) (a) (b) Figure 1. (a) Data for y and x plotted for various values of z and (b) a comparison of slopes for the 3 cases investigate. The log plot slope yields the vitamin B concentration. The slopes were shown to be significantly at the 90% confidence level, but the instructor ran out of time and did not include error bars. The slope changed as predicted by the Snirtenhoffer equation. Improvements to the lab might include advice on how to legally change my name to something less embarrassing. My whole life I have been forced to repeat and spell it. I really feel that this has affected my psychologically. This was perhaps the worst lab I have ever done in my academic career, primarily due to the fact that there was no lab time. I simply typed in this entire report and filled it with jibberish. Some might think nobody will notice, but I know that …… Harding reads every word. Acknowledgments The author acknowledges his elementary teacher for providing truly foundational instruction in addition and subtraction. Jenny Burninbalm was instrumental with guidance on use of the RT-345 dog scratching device. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 12

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