Mathematics 202 Spring 2011 +-------+----+----------+ | I | 20 | | Midterm Examination |-------+----+----------| | II | 20 | | _________________________________________ |-------+----+----------| (Student's Name) | III | 20 | | |-------+----+----------| Instructions: | IV | 20 | | |-------+----+----------| Read each question carefully, to be | V | 20 | | sure you understand what is being asked. |-------+----+----------| Show all of your work on these pages; use |=======================| the back of these sheets if you need more | Total | | space. In every case make sure that your | Score | | final answer is plainly shown. +------------+----------+ I. Among 56 computer science students who earned degrees in 2010, 25 had taken a graphics course, 23 had taken a database course, and 24 had taken an algorithms course; of these, the number who had taken both graphics and databases was 8, the number who had taken both databases and algorithms was 6, and the number who had taken both algorithms and graphics was 7. Moreover, two of of these strudents had taken all three of these subjects. (a) How many of these students took only the graphics course? Answer:________________ (b) How many students did not take any of these three courses? Answer:________________ (c) If a student did not take the algorithms course, what is the probability that this student also did not take the database course? Answer:________________ (d) If a student took more than one of these three classes, what is the probability that this student took graphics and algorithms? Answer:________________ Math 202 Midterm Second of Five Pages Name_______________ II. Three boxes contain varying numbers of colored marbles, as follows: Box A contains 5 red marbles and 5 white marbles Box B contains 6 white marbles and 4 red marbles Box C contains 3 blue marbles and 7 red marbles If an experiment is performed in which one marble, chosen at random, is removed from each of these boxes, then: (a) what is the probability that the three marbles selected all have different colors? (b) Given that the three marbles selected did indeed have different colors, what is the conditional probability that Box A was left with more red marbles than white marbles? Math 202 Midterm Third of Five Pages Name_______________ III. On her way to classes on the first day of a new semester, Sandra is briefly delayed by being stuck in the elevator. By the time she finally reaches the 5th Floor of Harney, classes have already started and the doors to the four classrooms are shut. At that point Sandra realizes she has forgotten the room-number of the class she is supposed to be attending, so she will have to open a classroom door, selected at random, and peek inside to see if she recognizes her professor; if not, she must try different doors, one at a time, until she finds the room she belongs in. Suppose X equals the number of wrong doors Sandra opens while trying to find her proper classroom. (a) List the possible values for this random variable X . (b) Show the probability distribution for X . (c) Compute the 'expected' number of wrong doors Sandra will have to open (i.e., the mean value for X ). (d) Compute the standard deviation for the variable X . Math 202 Midterm Fourth of Five Pages Name_______________ Table of areas under the standard normal curve from 0 to z z 0 1 2 3 4 5 6 7 8 9 0.0: 0.000 0.040 0.079 0.118 0.155 0.191 0.226 0.258 0.288 0.316 1.0: 0.341 0.364 0.385 0.403 0.419 0.433 0.445 0.455 0.464 0.471 2.0: 0.477 0.482 0.486 0.489 0.492 0.494 0.495 0.497 0.497 0.498 3.0: 0.499 0.499 0.499 0.500 0.500 0.500 0.500 0.500 0.500 0.500 IV. Whenver Bob takes his computer in for repairs, he doesn't know how many days he will have to leave it before it is ready to be picked up, but he believes that this unknown number X is a normally distributed random variable. Assuming X has a mean value of 8 days, with a standard deviation of 4 days, what is your estimate for Bob's chances of having his computer back within two weeks (or less) of taking it in? Math 202 Midterm Fifth of Five Pages Name_______________ V. According to Google's corporate website, its employees are allowed during their breaks to take gym classes in either yoga or dance. Suppose 70% of those who choose yoga during any given gym break will also choose yoga during their next gym break, and 80% of those who choose dance during any gym break will again choose dance on their next break. (a) Draw a transition diagram for this Markov Chain. (b) Show the transition matrix for this Markov Chain. (c) Assuming Pete went to dance class at yesterday's gym break, what is his probability of being back at the dance class during gym break tomorrow? (d) What percentage of the Google employees should we expect to find are attending the yoga class after a year's worth of working days have passed?