## 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

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## 1 BACKGROUND The new generation of enhanced mid core PICs such as the 16F1847 and the 12F1840 have an inbuilt temperature sensor. This sensor consists of a current source which flows through four diodes in series and the voltage drop across the diodes which is proportional to temperature can be measured by internally connecting the sensor to the ADC and determining the temperature based on the ADC value In this assignment the temperature sensor is used to create a simple thermometer application and to create an alarm should the sensor go outside the set value. Assignment Details 1) Determine the register settings needed to switch the sensor on and connect the temperature sensor to the ADC. Using appropriate values for Vref+ and Vref- display the ADC count value on the 7 segment display. 2) With reference to Microchip Application Note AN1333, “Use and Calibration of the Internal Temperature Indicator” (DS01333) determine an appropriate algorithm to convert from the ADC value to the temperature in degrees centigrade and implement it using a lookup table or otherwise. Display this value on the 7 segment display. Additional marks will be given for accuracy, calibration and averaging the temperature readings to give a more accurate, and a more stable temperature reading. . 2 In order to meet the specification the following will be required. i) Selection of appropriate microcontroller to meet the requirement of the task. ii) Development of an assembly language program to control the operation of the embedded system. iii) Thorough testing to ensure correct operation of the system. iv) Produce a project report to evidence all of the above. Follow Report Requirements (20 pages max) 1) Introduction – Clearly state the scope and aims and objectives of the project: Include Aims and Objectives, i.e. break down the project into smaller attainable aims and objectives for example one objective could be to develop a program to control the LED display. If all objectives are met then the overall project should have been completed. 2) Theory – Include any relevant theory 3) Procedure, Results Discussion – The report should show a methodical, systematic design approach. The microcontroller kits in the laboratory can be used as the hardware platform, however circuit diagrams should be included in the report and explanations of operation is expected. 4) Include flowcharts and detailed explanations of software development. Include appropriate simulation screen shots. Show and discuss results e.g. ADC program, LED program, etc. Include final/complete program. Were results as expected, do they compare favourably with simulated results, what could be done to improve the operation and accuracy of the system? 5) Conclusion – Reflect back on the original aims and objectives. Were they met if not why not? What further work could be carried out to meet aims and objectives etc? 3 Marks ALLOCATION Marks are allocated for the given activities as follows: MARK (%) PROJECT WORK 60 PROJECT REPORT 30 PRESENTATION MARK 10 ______ Total 100 The marks awarded for the microcontrollers in embedded system module will be made up as follows:- PROJECT MARK Have all of the specifications been met? Correct Register settings to switch on sensor and connect temperature sensor to ADC 5% Display two different characters on the 7 segment display 5% Display the ADC count value on the 7 segment display 10% Display the temperature on the seven segment display 20% Calibration 10% Accuraccy 10% Total 60% REPORT MARK Introduction and Theory 5% Procedure, Results and Discussion 20% Report Presentation 5% Total 30% PRESENTATION (POWER POINT) & DEMO Demonstration 10% Total 10% TOTAL 100% 4 Schematic for the Assignment Seven Segment Display Code ;************************************************ ;Appropriate values to illuminate a seven segment display ;with numbers 0 – 9 are extracted from a look up table ;and output on PORTB. ;A software delay is incorporated between displaying ;successive values so that they can be observed. ;(This program is useful demonstrating software delays, ; and look up tables. ; ;************************************************ ; list p=16F1937A #include

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