SECTION 1: Short-Answer Questions
Question 1 - Imagine that this assignment was a multiple-choice test with 10 questions, where each question had 5 choices (and only one right answer).
(a) If you were to guess at the answer to every question, what is the probability that you would answer at least one of the questions correctly?
(b) What assumptions did you make in calculating your answer in part (a) above?
Question 2 - Suppose a scammer is planning to set up a 'boiler room' to make phone calls to convince people that the Australian Tax Office is about to arrest them for tax evasion, and then to scare them into paying a 'fine' to avoid arrest. His past experience suggests that each phone will be in use half the time. For a given phone at a given time, let 0 indicate that the phone is available and let 1 indicate that a call is being made. Suppose that the scammer has set up 4 phones.
(a) Write out all the possible states of the four phones at any given time.
(b) Which outcomes would make up the event that exactly two phones are being used?
(c) Suppose the scammer had k phones. How many outcomes would allow for the possibility that at most one more call could be made?
Question 3 - Commuters leaving a train station can exit through any one of three turnstiles, A, B, or C. Assuming that a commuter is equally likely to select any one of the three turnstiles, what is the probability that among four commuters, two select gate A, and at least one selects gate C?
Question 4 - Let A and B be two events defined on a sample space S, and let |·| represent the number of outcomes in a given set. Furthermore, let |A?Bc| = 15, |Ac?B| = 50, |A?B| = 2, and |S| = 120. Sketch a Venn diagram showing the two sets and label the diagram with |A?Bc|, |Ac?B|, and |A?B|. How many outcomes belong to neither A nor B? How would you represent this quantity symbolically?
Question 5 - How many ways are there to permute the letters in the word STATISTICS?
Question 6 - A total of twelve hundred graduates of high schools in a large local government area in Perth have gotten into medical school in the past several years. Of that number, one thousand earned scores of 27 or higher on the GAMSAT (Graduate Medical School Admissions Test) and four hundred had GPAs (Grade Point Averages) that were 3.5 or higher. Moreover, three hundred had GAMSATs that were 27 or higher and GPAs that were 3.5 or higher. Sketch a Venn diagram to illustrate this problem. What proportion of those twelve hundred graduates got into medical school with a GAMSAT lower than 27 and a GPA below 3.5?
Question 7 - Suppose a single gene controls the color of hamsters: black (B) is dominant and brown (b) is recessive. Hence, a hamster will be black unless its genotype is bb. Two hamsters, each with genotype Bb, mate and produce a single offspring. The laws of genetic recombination state that each parent is equally likely to donate either of its two alleles (B or b), so the offspring is equally likely to be any of BB, Bb, bB, or bb (the middle two are genetically equivalent).
(a) What is the probability their offspring has black fur?
(b) Given that their offspring has black fur, what is the probability its genotype is Bb?
SECTION 2: Deep-Thought/Longer Questions
Question 1 - A child uses a home-made metal detector to look for valuable metallic objects on a beach. There is a fault in the machine which causes it to signal the presence of only 95% of the metallic objects over which it passes, and to signal the presence of 6% of the non-metallic objects over which it passes. Of the objects over which the machine passes, 20% are metallic.
(a) Find the probability that a given object over which the machine passes is metallic and the machine gives a signal.
(b) Find the probability of a signal being received by the boy for any given object over which the machine passes.
(c) Find the probability that the boy has found a metal object when he receives a signal.
(d) Given that 10% of metallic objects on the beach are valuable, find the proportion of objects, discovered by a signal from the detector, that are valuable.
Question 2 - A random sample of 6 items is taken from a large consignment and tested in two independent stages. The probability that an article will pass either stage is q. All six items are first tested at stage 1, and provided 5 or more pass, those which pass are retested at stage 2. The consignment is accepted if there is no more than one failure at each stage. Find expressions in terms of q for:
(a) The probability that stage 2 of the test will be required.
(b) The number of items expected to undergo stage 2.
(c) The probability P(q) of accepting the consignment.
(d) Show that dP/dq = 0 when q = 1 and find P(q = 0.9) and P (q = 0.8). Comment on your results.
Question 3 - A production facility employs 20 workers on the day shift (8:00 a.m.-4:00 p.m.), 15 workers on the swing shift (4:00 p.m.-midnight), and 10 workers on the graveyard shift (midnight-8:00 a.m.). A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45).
(a) How many selections result in all 6 workers coming from the day shift? What is the probability that all 6 selected workers will be from the day shift?
(b) What is the probability that all 6 selected workers will be from the same shift?
(c) What is the probability that at least two different shifts will be represented among the selected workers?
(d) Bonus Question: What is the probability that at least one of the shifts will not be represented in the sample of workers?
Question 4 - Brett and Margo have each thought about murdering their rich Uncle Basil in hopes of claiming their inheritance a bit early. Hoping to take advantage of Basil's predilection for immoderate desserts, Brett has put rat poison into the cherries flambé; Margo, unaware of Brett's activities, has laced the chocolate mousse with cyanide. Given the amounts likely to be eaten, the probability of the rat poison being fatal is 0.60; the cyanide, 0.90. And though Uncle Basil is a corpulent epicure, he's unlikely to drop dead for no reason at all.
Based on other dinners where Basil was presented with the same dessert options, we can assume that he has a 50% chance of asking for the cherries flambé, a 40% chance of ordering the chocolate mousse, and a 10% chance of skipping dessert altogether. No sooner are the dishes cleared away than Basil drops dead. Sherlock Holmes, who is asked to solve the crime, knows a thing or two about probability. To explain his reasoning to Dr Watson, Holmes begins by sketching a tree diagram-and you should too!-and then goes on to use probabilistic calculations based on the information given above, and using Bayes' rule, to identify the likely culprit. Who is it-Margo or Brett?
Question 5 - Airlines often overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function (pmf) of Y appears in the accompanying table. Answer all questions below using proper notation.
|
y
|
45
|
46
|
47
|
48
|
49
|
50
|
51
|
52
|
53
|
54
|
55
|
|
p(y)
|
0.05
|
0.1
|
0.12
|
0.14
|
0.25
|
0.17
|
0.06
|
0.05
|
0.03
|
0.02
|
0.01
|
(a) Why is this a legitimate pmf?
(b) Plot the pmf, and label it correctly.
(c) What is the probability that the flight will accommodate all ticketed passengers who show up? How would you write this probability?
(d) What is the probability that not all ticketed passengers who show up can be accommodated? How would you write this probability?
(e) If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list? How would you write these probabilities?
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