Lab Report Name Simple Harmonic motion Date: Objective or purpose: The main objective of this lab is to find the value of the spring constant (k) according to Hooke’s law. This lab also teaches us curve fitting and its application here in this lab.

Lab Report Name Simple Harmonic motion Date: Objective or purpose: The main objective of this lab is to find the value of the spring constant (k) according to Hooke’s law. This lab also teaches us curve fitting and its application here in this lab.

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Lab #03 Studying Beam Flexion Summary: Beams are fundamental structural elements used in a variety of engineering applications and have been studied for centuries. Beams can be assembled to create large structures that carry heavy loads, such as motor vehicle traffic. Beams are also used in micro- or nano-scale accelerometers to delicately measure and detect motions that trigger the deployment of an airbag. From a technical standpoint, a beam is a structure that supports transverse load. Transverse load is load that is perpendicular to the long axis of the beam. As a result, of transverse load, beams undergo bending, in which the beam develops a curvature. As the beam bends, material fibers along the beam’s long axis are forced to stretch or contract, which in turn causes a resistance to the bending. The fibers that are the farthest away from the center of the beam are forced to stretch or contract the most and thus, material at these extremities is the most important to resist bending and deflection. This topic is studied quantitatively in Strength of Materials (CE-303). Purpose: The purpose of this assignment is to accomplish the following goals: • Develop a simple experiment to achieve a goal. • Statistically and observationally analyze your data and interpret the results. • Summarize and present your data, results and interpretations. Procedure: 1. Working as a team, develop a procedure to carefully document the amount of bending a beam under-goes as loads are placed on it (this is your experimental protocol). You must select at least two different beam styles. 2. Collect the data points your experimental protocol calls for. You should conduct at least three trials and the order of data collection within those trials should be randomized. 3. Using the provided Excel deflection calculator, calculate the “predicted” deflection for each of the trials in your protocol. 4. Please observe the following MAXIMUM test torques to avoid damaging the beams. • Width Effect Beams: Small beam: 48 in-lbs, Medium beam: 80 in-lbs, Large beam: 120 in-lbs • Depth Effect Beams: Small beam: 8 in-lbs, Medium beam: 48 in-lbs, Large beam: 160 in-lbs Report and Presentation Requirements: 1. Title Page: Should include the title of the lab experiment, groups individual names (in alphabetical order by last name), data collection date, report due date, and course name and section. 2. Introduction: Briefly explain what you are trying to accomplish with this experiment. 3. Hypothesis Development: Should clearly state the three hypotheses, with respect to distance, beam size, and calculated versus actual deflection. Be sure to include logic to support your educated guess. 4. Method: Explain each activity performed during the data collection and analysis process. Provide a list of the equipment used and its purpose. 5. Analysis and Results: (1) Using the raw data, provide a table of descriptive statistics (mean, variance, and range) for each beam at each distance. (2) Provide a data table (average across 3 trials) showing the deflection for each beam at each distance. (3) Create one or more charts demonstrating the difference, if any, between the calculated and observed deflection for each beam. (4) Use the t-Test: Paired Two Sample for Means in Excel to determine if there is a statistically significant difference between predicted (calculated) deflection and actual (observed) deflection, assuming α = 0.05. Show the results for each beam. Note: To add in the Data Analysis package (under the data tab), go to Office Button -> Excel Options -> Add-Ins -> Manage Excel Add-Ins -> GO… -> check Analysis TookPak and click OK. For each table or chart, provide a description and explanation of what is being displayed. 6. Conclusions: Restate the hypotheses and explain whether or not the educated guess was correct. Include limitations of the experiment (in other words, describe other factors that would make the experiment better or possible errors associated with the experiment). Provide suggestions for future research. 7. Last Page: Include, at the end of the document, a summary of all the tasks required to complete the assignment, and which member or members of the group were principally responsible for completing those tasks. This should be in the form of a simple list. 8. Presentation: Summarize the report, excluding the last page. Due Date: This assignment is to be completed and turned in at the beginning of your laboratory meeting during the week of 11th March. Microsoft office package: Excel: Data tab functions, round, drag-drop, $-sign functions, Beginning of analysis toolpak-t-tests

Lab #03 Studying Beam Flexion Summary: Beams are fundamental structural elements used in a variety of engineering applications and have been studied for centuries. Beams can be assembled to create large structures that carry heavy loads, such as motor vehicle traffic. Beams are also used in micro- or nano-scale accelerometers to delicately measure and detect motions that trigger the deployment of an airbag. From a technical standpoint, a beam is a structure that supports transverse load. Transverse load is load that is perpendicular to the long axis of the beam. As a result, of transverse load, beams undergo bending, in which the beam develops a curvature. As the beam bends, material fibers along the beam’s long axis are forced to stretch or contract, which in turn causes a resistance to the bending. The fibers that are the farthest away from the center of the beam are forced to stretch or contract the most and thus, material at these extremities is the most important to resist bending and deflection. This topic is studied quantitatively in Strength of Materials (CE-303). Purpose: The purpose of this assignment is to accomplish the following goals: • Develop a simple experiment to achieve a goal. • Statistically and observationally analyze your data and interpret the results. • Summarize and present your data, results and interpretations. Procedure: 1. Working as a team, develop a procedure to carefully document the amount of bending a beam under-goes as loads are placed on it (this is your experimental protocol). You must select at least two different beam styles. 2. Collect the data points your experimental protocol calls for. You should conduct at least three trials and the order of data collection within those trials should be randomized. 3. Using the provided Excel deflection calculator, calculate the “predicted” deflection for each of the trials in your protocol. 4. Please observe the following MAXIMUM test torques to avoid damaging the beams. • Width Effect Beams: Small beam: 48 in-lbs, Medium beam: 80 in-lbs, Large beam: 120 in-lbs • Depth Effect Beams: Small beam: 8 in-lbs, Medium beam: 48 in-lbs, Large beam: 160 in-lbs Report and Presentation Requirements: 1. Title Page: Should include the title of the lab experiment, groups individual names (in alphabetical order by last name), data collection date, report due date, and course name and section. 2. Introduction: Briefly explain what you are trying to accomplish with this experiment. 3. Hypothesis Development: Should clearly state the three hypotheses, with respect to distance, beam size, and calculated versus actual deflection. Be sure to include logic to support your educated guess. 4. Method: Explain each activity performed during the data collection and analysis process. Provide a list of the equipment used and its purpose. 5. Analysis and Results: (1) Using the raw data, provide a table of descriptive statistics (mean, variance, and range) for each beam at each distance. (2) Provide a data table (average across 3 trials) showing the deflection for each beam at each distance. (3) Create one or more charts demonstrating the difference, if any, between the calculated and observed deflection for each beam. (4) Use the t-Test: Paired Two Sample for Means in Excel to determine if there is a statistically significant difference between predicted (calculated) deflection and actual (observed) deflection, assuming α = 0.05. Show the results for each beam. Note: To add in the Data Analysis package (under the data tab), go to Office Button -> Excel Options -> Add-Ins -> Manage Excel Add-Ins -> GO… -> check Analysis TookPak and click OK. For each table or chart, provide a description and explanation of what is being displayed. 6. Conclusions: Restate the hypotheses and explain whether or not the educated guess was correct. Include limitations of the experiment (in other words, describe other factors that would make the experiment better or possible errors associated with the experiment). Provide suggestions for future research. 7. Last Page: Include, at the end of the document, a summary of all the tasks required to complete the assignment, and which member or members of the group were principally responsible for completing those tasks. This should be in the form of a simple list. 8. Presentation: Summarize the report, excluding the last page. Due Date: This assignment is to be completed and turned in at the beginning of your laboratory meeting during the week of 11th March. Microsoft office package: Excel: Data tab functions, round, drag-drop, $-sign functions, Beginning of analysis toolpak-t-tests

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In an experiment, a scientist fed radioactive nucleotides to a cell that was previously non-radioactive. After the cell duplicated its DNA, which statement would be true?

In an experiment, a scientist fed radioactive nucleotides to a cell that was previously non-radioactive. After the cell duplicated its DNA, which statement would be true?

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Physics Lab 1 Projectile Motion We will use this 2-D “Golf Simulation” to explore combined horizontal and vertical motion (select experiment #7). This simulation provides a fun-filled way to examine 2-D projectile motion with and without air resistance. Before to start the experiment, take a few minutes playing with the simulation. Adjust the initial velocity by adjusting the launch speed and launch angle. See how many adjustments you have to make in order to get a hole-in-one. Turn the air on and off, turn the trails on and off, notice the time, and notice the shape of the curves. Instructions: • Go to http://www.physicslessons.com/exp7b.htm • Set the launch velocity to 60 m/s, trail “on” and “no air”. • Change the launch angle to 15 degree, click the launch button and take note of the horizontal displacement x. Repeat the experiment (changing the angle) and fill the first table (at the left). • Click the “no air” button (so it changes to “air”), repeat the experiments and fill the second table (at the right). Displacement [without air] Displacement [with air] Set launch speed, Vo = 60 m/s Set launch speed, Vo = 60 m/s Angle,  (deg) x (m) Angle,  (deg) H-Dis, x (m) 15 15 25 25 35 35 40 40 43 43 45 45 47 47 50 50 55 55 65 65 75 75 Questions: 1. What angle corresponds to the greatest horizontal range for the “without air” condition? What angle corresponds to the greatest horizontal range for the “with air” condition? Why is there a difference? 2. Describe the difference between the general shape of the trails for the two separate cases. 3. Do you notice any symmetry between high and low angles for either case? Describe the symmetry. 4. When practicing (playing) with the simulation earlier, how many tries did it typically take you to land the ball in the hole?

Physics Lab 1 Projectile Motion We will use this 2-D “Golf Simulation” to explore combined horizontal and vertical motion (select experiment #7). This simulation provides a fun-filled way to examine 2-D projectile motion with and without air resistance. Before to start the experiment, take a few minutes playing with the simulation. Adjust the initial velocity by adjusting the launch speed and launch angle. See how many adjustments you have to make in order to get a hole-in-one. Turn the air on and off, turn the trails on and off, notice the time, and notice the shape of the curves. Instructions: • Go to http://www.physicslessons.com/exp7b.htm • Set the launch velocity to 60 m/s, trail “on” and “no air”. • Change the launch angle to 15 degree, click the launch button and take note of the horizontal displacement x. Repeat the experiment (changing the angle) and fill the first table (at the left). • Click the “no air” button (so it changes to “air”), repeat the experiments and fill the second table (at the right). Displacement [without air] Displacement [with air] Set launch speed, Vo = 60 m/s Set launch speed, Vo = 60 m/s Angle,  (deg) x (m) Angle,  (deg) H-Dis, x (m) 15 15 25 25 35 35 40 40 43 43 45 45 47 47 50 50 55 55 65 65 75 75 Questions: 1. What angle corresponds to the greatest horizontal range for the “without air” condition? What angle corresponds to the greatest horizontal range for the “with air” condition? Why is there a difference? 2. Describe the difference between the general shape of the trails for the two separate cases. 3. Do you notice any symmetry between high and low angles for either case? Describe the symmetry. 4. When practicing (playing) with the simulation earlier, how many tries did it typically take you to land the ball in the hole?

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Lab Report : The purpose of this experiment is to learn about uncertainty in measurement, and how to perform calculations with those uncertainties. The calculation of the density of a wooden block was used as an example.

Lab Report : The purpose of this experiment is to learn about uncertainty in measurement, and how to perform calculations with those uncertainties. The calculation of the density of a wooden block was used as an example.

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Materials Chemistry for Engineers 1. In the van der Waals corrections to the Ideal Gas Law: (P + a/V2)(V – b) = nRT (a) What do a and b correct for from the Ideal Gas Law? (b) How would one determine a and b experimentally? Describe a proposed experiment and data analysis method for your experiment. 2. (a) What are the assumptions of the Ideal Gas Law? How did van der Waal modify these assumptions to come up with his equation of state? (b) what is an equation of state, in general? Describe in your own words. 3. Given the following data: Material a b_____ (l2.atm/mole2) (l/mole) N2 1.39 0.03913 NH3 4.17 0.03107 Aniline 26.50 0.1369 Benzene 18.00 0.1154 (a) Plot P vs. T for each gas using the van der Waals equation of state. Assume that you have a 1 liter volume and 1 mole of gas and plot the temperature on the x-axis from room temperature to 1400 K (pressures should range from about 0 atm to about 120 atm, depending on the gas). Plot the Ideal Gas Law with the other data on one plot. Are the interactions between molecules attractive or repulsive at low temperature? How do you know? What is happening with the gases at high temperature? Is one of the gases different from the others at 1400 K? (b) Discuss the nature of the intermolecular interaction that creates the deviation from ideality for each material. Are there induced dipole-induced dipole interactions, iondipole interactions, etc. for each of the different gases? Draw their chemical structures. 4. Ethane (CH3CH3) and fluoromethane (CH3F) have the same number of electrons and are essentially the same size. However, ethane has a boiling point of 184.5 K and fluoromethane has a boiling point of 194.7 K. Explain this 10 degree difference in boiling point in terms of the van der Waals forces present. Bonus, what is the size of each molecule? Show your calculation/sources.

Materials Chemistry for Engineers 1. In the van der Waals corrections to the Ideal Gas Law: (P + a/V2)(V – b) = nRT (a) What do a and b correct for from the Ideal Gas Law? (b) How would one determine a and b experimentally? Describe a proposed experiment and data analysis method for your experiment. 2. (a) What are the assumptions of the Ideal Gas Law? How did van der Waal modify these assumptions to come up with his equation of state? (b) what is an equation of state, in general? Describe in your own words. 3. Given the following data: Material a b_____ (l2.atm/mole2) (l/mole) N2 1.39 0.03913 NH3 4.17 0.03107 Aniline 26.50 0.1369 Benzene 18.00 0.1154 (a) Plot P vs. T for each gas using the van der Waals equation of state. Assume that you have a 1 liter volume and 1 mole of gas and plot the temperature on the x-axis from room temperature to 1400 K (pressures should range from about 0 atm to about 120 atm, depending on the gas). Plot the Ideal Gas Law with the other data on one plot. Are the interactions between molecules attractive or repulsive at low temperature? How do you know? What is happening with the gases at high temperature? Is one of the gases different from the others at 1400 K? (b) Discuss the nature of the intermolecular interaction that creates the deviation from ideality for each material. Are there induced dipole-induced dipole interactions, iondipole interactions, etc. for each of the different gases? Draw their chemical structures. 4. Ethane (CH3CH3) and fluoromethane (CH3F) have the same number of electrons and are essentially the same size. However, ethane has a boiling point of 184.5 K and fluoromethane has a boiling point of 194.7 K. Explain this 10 degree difference in boiling point in terms of the van der Waals forces present. Bonus, what is the size of each molecule? Show your calculation/sources.

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QUESTIONS ! 1.State what the experiment checks. !!!!!!!!! 2. (a) How does the centripetal force vary with the speed of rotation for a constant radius of the path? ! !!!!!!!! (b) How does it vary with the radius of the path for a constant speed of rotation? !!!!!!! 3. Distinguish between centripetal force and centrifugal force. Explain in what direction each force is acting and on what it is acting. !!!!!!!! 4.Calculate at what speed Earth would have to rotate in order that objects at the equator would have no weight. Assume the radius of Earth to be 6400 km. What would be the linear speed of a point on the equator? What would be the length of a day (time from sunrise to sunset) under these conditions? !!!!!!!! 5. Engines for propeller-driven aircraft are limited in their maximum rotational speed by the fact that the tip speed of the propeller must not approach the speed of sound in air (Mach I). Taking 6 ft as a typical diameter for a propeller of a light airplane and 1100 fils as the speed of sound, find the upper limit on the rpm (revolutions per minute) of the propeller shaft.

QUESTIONS ! 1.State what the experiment checks. !!!!!!!!! 2. (a) How does the centripetal force vary with the speed of rotation for a constant radius of the path? ! !!!!!!!! (b) How does it vary with the radius of the path for a constant speed of rotation? !!!!!!! 3. Distinguish between centripetal force and centrifugal force. Explain in what direction each force is acting and on what it is acting. !!!!!!!! 4.Calculate at what speed Earth would have to rotate in order that objects at the equator would have no weight. Assume the radius of Earth to be 6400 km. What would be the linear speed of a point on the equator? What would be the length of a day (time from sunrise to sunset) under these conditions? !!!!!!!! 5. Engines for propeller-driven aircraft are limited in their maximum rotational speed by the fact that the tip speed of the propeller must not approach the speed of sound in air (Mach I). Taking 6 ft as a typical diameter for a propeller of a light airplane and 1100 fils as the speed of sound, find the upper limit on the rpm (revolutions per minute) of the propeller shaft.

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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<D≪N. What is the sample space? If we are interested in knowing the number (count) of defective parts in the sample space, describe how, the concept of a random variable could help. Define a random variable Y and describe what values the random variable Y can take? What does it mean if we say Y=5? Q17. Suppose an experiment consists of tossing two fair coins. Let Y denote the number of heads appearing. 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=1? What are the probabilities associated with each outcome? What is the sum of the probabilities associated with all possible values that Y can take? Q18. A lot, consisting of 100 fuses, is inspected by the following procedure. Five fuses are chosen at random and tested: if all 5 fuses pass the inspection, the lot is accepted. Suppose that the lot contains 20 defective fuses. What is the probability of accepting the lot? Define the random variable, its purpose, and the formula/concept that you would use. Q19. In a small pond there are 50 fish, 10 of which have been tagged. If a fisherman’s catch consists of 7 fish, selected at random and without replacement. Give an example of a random variable that can be defined if we are interested in knowing the number of tagged fish that are caught? What is the probability that exactly 2 tagged fish are caught? Define the random variable, its purpose, and the formula/concept that you would use. Applied to Quality Control Q20. My manufacturing firm makes 100 cars every day out of which 10 are defective; the quality control inspector tests drives 5 different cars. Based on the sample, the quality control inspector will make a generalization about the whole batch of 100 cars that I have on that day. Let d denote the number of defective cars in the sample What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective- call this P(d=0); (b) 20% of the cars are defective- call this P(d=1); (c) 40% of the cars are defective- call this P(d=2); (d) 60% of the cars are defective- call this P(d=3),(e) 80% of the cars are defective- call this P(d=4), and (f) 100% of the cars are defective- call this P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Q21. Assume that the quality control inspector is selecting 1 car at a time and the car that he tested is put back in the pool of possible cars that he can test (sample with replacement). Let d denote the number of defective cars in the sample (n) What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective, (b) 20% of the cars are defective, (c) 40% of the cars are defective, (d) 60% of the cars are defective, (e) 80% of the cars are defective, and (f) 100% of the cars are defective. Let’s call these P(d=0)….P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Interesting Problems Q22. A closet contains n pairs of shoes. If 2r shoes are chosen at random (2r<n), what is the probability that there will be no matching pair in the sample? Q23. In a draft lottery containing the 366 days of the leap year, what is the probability that the first 180 days drawn (without replacement) are evenly distributed among the 12 months? What is the probability that the first 30 days drawn contain none from September? Q25. You and I play a coin-tossing game. If the coin falls heads I score one, if tails, you score one. In the beginning, the score is zero. What is the probability that after 2n throws our scores are equal? What is the probability that after 2n+1 throws my score is three more than yours?

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