Homework assignments, Probability & Statistics (Math 266) Spring 2013
Prof. Janeba, Spring 2013
Homework assignments for the week of:
Jan. 14  Jan. 21  Jan. 28  Feb. 4  Feb. 11  Feb. 18  Feb. 25  Mar. 4  Mar. 11  Mar. 18  Apr. 1  Apr. 8  Apr. 15  Apr. 22  Apr. 29
Final Exam
Note: The dates below are the dates the problems are assigned. Problems should be considered "due" by the next class period after the assignment date, unless otherwise noted.
Week #1:
 Jan. 14:
 Read course syllabus (really, there could be a pop quiz on Wed.)
 Read the text, sections 1.1 and 1.2
 Start making your list of terminology, definitions, and theorems, since they aren't highlighted in the text
 Look at Prof. Janeba's
survival guide
to get important advice about reading a text. This survival guide
is written for calculus students, but it all applies to our class as well.
 Problem 1: Roll three (sixsided, "normal") fair dice.
What's the chance the sum of the three dice is exactly 5? (There are
several approaches to doing this that aren't terribly hard).
 Problem 2: How many ways are there to draw 50 M&Ms
and get exactly one green (when we only distinguish green from
nongreen, i.e. blue and orange are treated as the same result 
nongreen)? How about exactly 2 greens?
 Also, do p. 15 #1.1, 1.2, 1.5, 1.6, 1.9, and p. 19 #1.27 .
 Come to class on Wednesday ready to discuss; bring your reading notes.
 Jan. 16:
 Read text section 1.3 and 1.4, and review sigma notation by reading appendix A.1 (we'll get to A.2 later)
 Do text problems:
 theoretical: 1.10, 1.11, 1.12, 1.14, and 1.17
 applied: 1.31, 1.33, 1.34,
1.36, 1.39, 1.43, 1.45, 1.47 (the answer
key for 1.47a appears to be high by 100)
 Jan. 18:
 Read text sections 2.12.4
 Do text problems (these are spread throughout chapter 2, but
they aren't hard to find if you look. For example, 2.5 is on
p.38):
2.1,2.3,2.4,2.5, 2.35,2.36,2.37,2.39,2.41,2.43,2.47,2.51, 2.53
Week #2:
 Jan. 21: Read chapter 2.5. Do: catch up on your
prior homework, come to my office tomorrow with questions if you have
some, and...
 Also do these: 1) Explain in words and using a simple Venn diagram why, if A and B are sets in some sample space S, then (AB')(AB) = A.
 Use #1 above as a lemma to do 2.8
 Do 2.9,2.15, 2.54, 2.55, 2.57, 2.59, 2.69
 Jan. 23: Quiz #1
 Read chapter 2.6,2.7
 Do 2.17, 2.21, 2.23, 2.76,2.79,2.89, 2.93
 Jan. 25:
 Read 2.8
 Do these problems (the questions are long, but the answers aren't proportionally long).
 An ordinary deck of cards is shuffled well, and two cards are dealt without replacement. Let A be the event that the first card is an ace. Let B be the event that the second card is an ace.
 Compute P(B), P(A), P(BA) and P(BA') by
reasoning verbally about the cards. You don't need to give any
extensive combinatorial tree argument here unless you really want to.
 Compute P(BA) and P(BA') by using results from the previous part and using the formula for conditional probability in a "backwards" direction.
 Given that B = (BA)(BA'), compute P(B) from P(BA) and P(BA').
 Finally, for your computation of B in the previous step, explain what the individual P's in your formula mean, in terms of cards and common sense, to someone who doesn't understand formal probability.
Use language like "the chance that <this thing> happens if <that thing> has already happened.
 Drug testing in professional cycling is in the news these
days, with the assertion that drug use was and or is rampant in the
sport. However, around 2001 or so, it was commonly asserted that
such cheating was rare. We investigate a hypothetical world in
which such cheating was, in fact, rare.
 Suppose that there are 5000 athletes active in a sport, of whom (though no one knows it), 200 are using drug X.
 Suppose that the test used by authorities to detect the use of drug
X correctly identifies 97% of drug users who are tested (so 3% of those
using the drug get a negative test result  the test reports "not
using").
 Suppose further that the same test correctly identifies 99%
of nonusers, i.e. reports 99% of them as nonusers, and 1% as
users. We say these 1% have gotten a "false positive"
result. Positive because the test identified them as users,
"false" because they were in fact not users.
 Let A be the event that someone is using the drug, and B be the event that they test positive (correctly or falsely).
 Explain what P(BA) and P(BA') mean in practical terms for the athletes  give the numbers AND explain what those numbers are telling us.
 Now explain what P(AB) means in practical terms for the athletes, again with numbers AND interpretation of those numbers.
 This example is not as stark as the one in Friday's class
session  I have changed the percentages to make them less troubling,
but there are still significant implications. Write a sentence or
two about their practical meaning for drug testing in this sport.
 Do text problems 2.95, 2.97 (you may assume the town is so
large that nonreplacement does not significantly change the odds of
selecting someone with a deficiency), 2.99, 2.109
Week #3:
 Jan. 28:
 Read 3.1,3.2
 Do these problems:
 (A repeat from class to get you started) Give a single
formula for the probability of getting at least one ace in six rolls of
a fair, sixsided die.
 Now suppose you have a 12sided fair die, which has an "ace"
on exactly one side. Give a formula for the probability of
getting at least one ace in 12 rolls of this die.
 A fair spinner has 60 possible outcomes, all equally likely,
and exactly one of the outcomes is "1". Give a formula for the
probability of getting at least one "1" in 60 spins of this spinner.
 One unusual way to define the constant e is that e is the limit, as n goes to infinity, of (11/n)^{n}. Use this to note a similarity in all the answers above.
 Do 2.104 (note we are picking the labor disputes at random),
2.105, 2.106 (here we are picking onecar accidents at random), 2.109
(here we are finding probabilities for randomly selected orders).
 Hint for above: the rule of total probability and Bayes' Theorem. Some answers
 Do text problems 3.1, 3.4
 Jan. 30: Quiz #2
 Read 3.3, 3.4 in our text. Note:
 3.3 appears to be pretty simple, but there are some nonobvious
consequences of our 3.3 definitions that show up in 3.4. There
are many counterintuitive points in 3.4, so be sure to read 3.4
carefully.
 Review of discrete random variables:
 Do 3.3,3.4,3.9 (note 3.9 is about F, not f, remembering what I said about casesensitive notation), 3.11 (note 3.11 is about a distribution function F, not a probability distribution f), 3.13, 3.85, 3.87,
 Problems for continuous random variables:
 Do 3.16, 3.17, 3.19 (this requires an improper integral), 3.20, 3.21, 3.29, 3.91, 3.94
 Feb. 1:
 Today's bonus problem  if you wish, you may turn this in on
Monday for extra credit: Suppose you have a specific pair of
events A and B living in some sample space S. An attribute that the pair of events may or may not have is that they are (or aren't) mutually exclusive, a.k.a. disjoint. Note that this is not an attribute that one single event could possibly have, it only makes sense to ask about disjointness when you have a pair of events. Likewise, our pair of events may or may not be independent. Our questions are these: For our specific pair A,B, what combinations of the attributes mutally exclusive and independent
are possible? e.g. can both happen with the same pair?
neither? either one without the other? If you say a
combination is possible, give an example of two events that have this
combination of attributes. If you say a combination is not
possible, explain why. Note that you can generate all needed
examples from coin tosses and dealing cards from a shuffled deck.
 Reread 3.3, 3.4, and catch up on the homework already given.
Week #4:
 Feb. 4: Review! Catch up!
 Do 3.95 with this correction: In part (b), it is the daily capacity of this city's water supply that is given. Note you can use Wolfram Alpha or most calculators to find the integrals here.
 Do 3.96, but before you do the calculations, tell us why the part "F(x) = 0 for x5" is not only plausible, but necessary, given the wording of the problem. Hint: What does F(4) mean in the context of this problem?
 3.96 is relevant to the concept of life expectancy. For the US, in 2009 the life expectancy for females was 80.9 years per the CDC,
(76.0 for males), but in the same year, life expectancy for 65yearold
women in the US was 20.3, i.e. they were "expected" to live another
20.3 years, to the age of 85.3, (17.6 and 82.6 for males). How
can this even make sense? (Very easily, it turns out).
 I have added some answers to chapter 3 on this page, including some tips on using calculators for integrals.
 Feb. 6: Exam #1
 Read 4.1,4.2 (we will come back to the rest of chapter 3 later)
 Do exercises 4.1, 4.61, and 4.63
 Feb. 8:
 Do exercises 4.2, 4.3, (hint: modify the proof for the other case that is given in the text) 4.7, 4.9, 4.65
 Let X be the random variable that equals the number of flips of a fair coin required to "get a head."
 Write out the first several terms of the probability distribution for X, then figure the general pattern.
 Verify that the sum of the values in the prior step is indeed 1. If you need help with the infinite sum, see Things You Need To Know.
 Write out the infinite sum that gives the expected value of X. Add up the first 68 terms and compare to your intuition about this expected value.
 We will look at an exact way to evaluate this sum on Monday.
Week #5:
 Feb. 11: Read 4.3,4.4
 For each of the following datasets, give the mean and standard deviation .
 Remember that we can compute these by thinking of X as the result of a random selection from the dataset. Then and are as described in 4.3.
 {1,2,3,4,5,6,7}
 {0,0,0,2,2,2}
 {5,5,5,5}
 {1,2,3,4,5,5,6,7,12}
 For each of the datasets above, using the values of and found above, verify that Chebyshev's theorem holds for k = 1,2,3 (See chapter 4.4). (Write out what Chebyshev's theorem says in each case for each of k=1,2,3 and verify that it occurs).
 Some answers
 Do 4.17 (in the continuous random variable case), 4.19, 4.23, 4.26 (just go ahead and use the definitions for _{3} and ), 4.31, 4.69, 4.73 [Answer for 4.26b is 0.1718]
 Feb. 13: Quiz 3
 Read 4.9. Note that the sample standard deviation formula
is different from the one we have discussed; our SD formula is for
populations; we will discuss the distinction later.
 Do 4.32, 4.62, 4.64,
4.71, 4.74 (the answer key for 4.75 is actually for 4.74), 4.75 (let's
assume the time units are minutes; the answer key has no answer here).
 Feb. 15:
Week #6:
 Feb. 18:
 Read 3.5 (remember, we said we'd go back and pick these up)
 Do 4.34, 4.37, 4.38 (for 4.37, integrate from  to 0 and 0 to separately. For 4.38, use the series for 1/(1a) and plug in something clever)
 Work on project.
 Feb. 20: Quiz 4
 Feb. 22: Project #1 due, 4:00 p.m.
 Do problems 3.42, 3.43, 3.46, 3.47 (note 3.47 is asking for the cumulative distribution function),
 If the joint probability function for f is given by f(x,y)=kx(yx) for 0<x<1 0<y<2 and f(x,y)=0 elsewhere, find what k must be.
 For the previous problem, find the cumulative distribution function F(x,y). Check your answer by using your formula for F to compute f and verifying you get back to where we started.
 Do 3.54, 3.55, 3.58,
 Here are some answers.
 Read 3.6 & 3.7
Week #7:
 Feb. 25:
 Read 3.8 (stemandleaf, histograms)
 Do 3.93, 3.97, 3.99, 3.101, 3.105 (some of these are review),
3.111, 3.116 (this last one is a bit tedious, but we should do at least
one of them)
 Feb. 27: Quiz 5
 Read sections 5.15.4 (5.15.3 totals less than 2 pages; note carefully in 5.3 the restriction that x=0,1, i.e. only two values; the meat is in 5.4)
 Do problem 3.127, with the change that you make an actual histogram as described in class (area = percentage) and use at least three different widths among your class intervals, just for practice.
 Do problems 5.1 (see appendix A.2 for some useful formulas),
5.2 (skip the part about the limit of the derivative, it's ugly),
5.5ab, 5.41, 5.43
 Mar.
1:
 Do problems 5.8, 5.9 (Hint: If I multiply the probability
by a number larger than 1, the probability goes up. Otherwise it
goes down.)
 Also problems 5.42 (answer .2066), 5.43, 5.44 (asnwers a: .16669, b: 0.4073, c: also 0.4073, 5.45, 5.51
 Read 5.55.6; focus on what each distribution is for, or what it tells us.
 Do 5.57, 5.59, 5.61, 5.63
Week #8:
 Mar. 4:
 Do problem 5.54, 5.50, and use this web page (or google "cumulative binomial calculator") to do 5.52, 5.53
 Do problems 5.64,5.66 hint: Use the hypergeometric
distribution, 5.69 (see example 5.7 and the discussion immediately
preceeding it).
 Make up a summary of the material we have learned: What
are the important concepts, what distinguishing features help you tell
a problem of one sort from a problem of another sort, what important
tools do we have to solve these problems?
 (See A Math Course Survival Guide.
This one was written for freshman calculus, but 95% of it applies to
P/Stat, particularly the parts about how to read the text and what you
must get out of it.)
 Q&A session Tuesday evening at 7; location TBA. Bring questions, or we won't have anything to talk about.
 Mar. 6: Exam 2
 Read chapter 5.7 on the Poisson distribution
 Take the stoplight problem from your exam answer key  can you recalculate the answer to problem 4 using a Poisson distribution? Hint: is the expected number of accidents; x is the actual number of accidents, which may or may not be different.
 Do 5.71, 5.73, 5.75, 5.77 (what is the significance of =3.3?
Can you give an everyday interpretation?) (Note  these problems
should be quick. If not, reread the section)
 At the fishin' hole, we catch on average 3 fish per
afternoon. What is the probability that we catch 7 fish?
(Hint: If this were a binomial problem, our expected number of catches
would be n.
If we think of every instant of time that we fish as a trial that
either succeeds or fails, then we have infinitely many trials, and the
probability of success on any one is zero  how are these hints
relevant to this section of the text?)
 Mar. 8:
 A certain stretch of highway averages 2 accidents per
week. Assuming the chance of an accident at any instant is
independent of any other instant, use what you know to find the chance
of 4 accidents in a week. For the random variable X=number
of accidents in one week on this stretch of highway, the mean is ___
and the variance is ___. Combine these two numbers in a
onesentence nontechnical statement that you could publish in the
newspaper (i.e. what do they mean?)
 Do 5.53 (Hint: look at what we did with the binomial distribution), 5.81
 Read 6.1, 6.2, and 6.5 (we are jumping around a bit). Do 6.31, 6.36, 6.62, 6.63 (hint: Use tables for the last two)
Week #9:
 Mar. 11:
 Exam 2 optional extracredit opportunity: Today in class
we derived the momentgenerating function for the standard normal
distribution; it was M_{X}(t) = e^{t²/}^{2}^{}. Expand this MGF with a Maclaurin series to find and for the standard normal distribution. Show work in some detail.
 Due (if you choose to do it at all) at the start of class on Wednesday.
 Problem: We transformed the standard normal probability density function, f(x) = 1/√(2)*e^{x²/}^{2},
into the "general" normal f(x;µ,) = 1/(√(2))*e^{}^{(}^{x}^{µ)}^{²/}^{(2}^{²)}.
Show that the integral of the latter over the entire real line is equal
to 1 by using a clever substitution to transform the latter integral
into the former. Note explicitly one or two places where we use the fact that is positive.
 Table III in the back of our textbook (p. 574) is rather
cryptic. Here is a fact to help you decrypt the table: the
area under the standard normal pdf from 0 to 1.96 is 0.4750.
 Find the numbers above in the table & find how they relate,
Now use the table (because you ought to know how to use a table) to find the areas under the standard normal pdf as indicated:
 between 0 and 0.85
 between 0 and 2.17
 between 1 and 2 (you'll need to use two table entries).
 between 1.5 and +1.5 (hint: Then normal pdf is symmetric about x=0)
 between 1.2 and +1.7
 below 0.73
 below 0.21
 above 1.57
 If you have a calculator with a normal pdf/cdf built in, learn its syntax to confirm some of the answers above.
 Another random variable W has a normal distribution with mean 42 and standard deviation 4. Find each.
 Hint: First "standardize" W to Z by the transformation Z = (Wµ)/, then use the normal table or calculator.
 P(42<W<46)
 P(42<W<49)
 P(38<W<46)
 P(W<45)
 P(43<W)
 Do problem 6.67, 6.71
 Some answers
 Mar. 13: Quiz 6
 Read section 6.6, especially example 6.6, giving attention to
that ".5" continuity correction business, and do 6.31, 6.77, 6.79, then
6.75 (in that order), and 6.80
 Mar. 15:
 Read section 8.1,8.2, and 8.3; ask particularly what question 8.3 addresses that is different from 8.2
 Do 8.2, 8.3, 8.4 (theoretical) and 8.61, 8.63, 8.64, 8.69, then 8.58, 8.59, 8.62
Week #10:
 Mar. 18:
 Our reading points up the oftrepeated rule of thumb, that if n>30, then X_{bar}'s distribution may be approximated as normal, although if X's distribution is already normal, X_{bar} 's will be as well, for any value of n.
Suppose that X_{1},X_{2}, and X_{3}, have a uniform distribution on [0,1] (see section 6.2). Thus each X_{i} takes on values between 0 and 1.
 What is a possible range of values for X_{bar} in this context?
 Using the information about the uniform distribution in 6.2 and what we read about X_{bar} in 8.2,
show that the criterion in Prof. Janeba's rule of thumb, "X_{bar}'s distribution may be approximated as normal,
if [mean] 2 to 3 SD's makes sense in the given context," is just barely
met here. (Show the criterion is met, you need not show that the normal approximation is good;
it is.)
 According
to HANES, adult males aged 1824 in the US have average height 70" with
SD 2.5" (the values are approximate, assume for this problem they are
exact).
 Then for a random sample of 400 of these males, the average will be _____ (mean for X_{bar}) give or take ______ (SD for X_{bar}) or so.
 What is the chance that the sample average will be over 70.5"? How about 70.1"?
 (Reread problem 8.5, then do this:) A hypothetical population
is 55% female and quite large. A random sample of 144 is taken
(far less than 5% of the population).
 The proportion in the sample that are female will be _____ (mean for Capital Theta_{hat}), give or take ______ (SD for Capital Theta_{hat}) or so.
 What is the chance that the sample proportion will be under 50%?
 Some answers
 Finally, do 8.67 and 8.73.
 Mar. 20: Quiz 7
 Mar. 22:
 Read 11.1 (half a page), 11.2, and 11.3 (we are again jumping around)
 Think about how confidence is not the same as probability, i.e. why being 95% confident that an error is less than some number is not the same as there being a 95% probability that the error is less than that number.
 Do 11.21, 11.23, 11.25,11.29 (solve directly, OR use the referredto exercise 11.6), then do 11.4, 11.5, 11.6
 Note: I have proofread the answers in the back of the book for these; they are correct.
 If you have not yet submitted the required Group Evaluation for project 1, do so before midweek of break. There may still be a few of you who need to do this.
 There will be a 5% penalty assessed on the project 1 score of
students who do not submit an evaluation. It's quick, it's easy, do it. See the project assignment for details.
 Submit preferences for your group partners for project 2 by
Monday, April 1. Just email me a short note saying
who you would like to work with.
Spring Break Mar. 25  29
Week #11:
 Apr. 1:
 Read 11.4, do #11.38, 11.39, 11.41, 11.43, then 11.12.
 Hint for 11.12: recall what we said about random variables X whose values can only be 0 or 1: _{x} can be at most 1/2; and we have a formula for _{x} that has thetas in it.
 Added: Read this bit that I showed in class.
 Apr. 3: Quiz 8; Project 2 assigned
 Catch up on confidence interval homework & work on project
 A shortessay question: We use the terms "probability,"
"chance," and "likelihood" pretty much interchangeably  but
"confidence" apparently means something different. What is the
difference between saying:
 "I am 95% confident that <something> is true" and
 "There is a 95% chance that <something> will happen" ?
 This is a question you should be prepared to answer on the
next exam, along with the "what does 95% confidence mean?" as outlined
in this web page.
 Apr. 5:
 Read chapter 13.113.3
 Do problems 13.19, 13.20, 13.21, 13.25, 13.27, 13.33 (this last exercise illustrates an important point), 13.39
 Looking more at class notes than at definition 13.1 (which is
true but IMHO convoluted), write a 23 sentence explanation for the
layperson of "what a P value means." This may help; on that page, you may substitute "random variable" for "box" or "box model."
Week #12:
 Apr. 8:
 Reread p. 405, and do 13.29,13.30, & 13.31
 Do 13.57 & 13.59 also; compare the example from class today, or read section 13.5
 Work on your project. DO come see me if you're having trouble figuring this out.
 Apr. 10: NO quiz
 Do 13.37, 13.39, 13.41 (well, you can use the table to give a range for P), 13.43
 Read 13.5, and do 13.57 & 13.59 if you haven't already; also 13.61, 13.65
 Apr. 12:
 Make yourself an outline of the topics we have covered since the last exam. (There are quite a few!)
 Here are just a few of the things you'll want to consider:
 What kinds of Standard Deviations / Standard Errors have we studied? (there are several)
 What does the Central Limit Theorem tell us (and not tell
us)? What are various rules of thumb for when the normal
approximation applies?
 For each of the various hypothesistest formats we have
studied, what requirements do we have for each one before we can
"safely" use it?
 Some questions to consider:
 Which makes us more likely to reject H_{o}  a large P value or a small one? Alternately,
 If we agree to test at the =0.05 level, then we reject H_{o} if P is (pick one: larger, smaller) than 0.05.
 If we reject H_{o} when P is 0.035, would we be more or less confident about our rejection if P were instead 0.04?
 What does "95% confidence" mean? (Or 99%, or 80%)? See the web pages cited on April 3 for some help.
 When we compute P for a hypothesis test, what is it the probability of? Warning: It is NOT the probability that H_{o} is true. H_{o} is either true or false (in almost all our tests), and is not subject to probability. One hint for both P and confidence intervals: probabilities in these contexts are probabilities that samples come out a certain way.
 Mix up a bunch of the homework problems we have already assigned, and try to tell which method each one requires.
 DO a bunch of specific homework problems.
 Watch this space to see if more review items appear on Sunday.
Week #13:
 Apr. 15: Exam 3
 Read chapter 13.6 & 13.7;
 Do 13.8 (just decipher the f_{i,j} and e_{i,j} notation using the definitions in the text, and the fact that one of the sums has a double sigma, then it's just some algebra)
 and 13.14 (again, my main interest here is that you decipher what the question is asking by learning the notation).
 Apr. 17: Student Scholarship Recognition Day (class does not
meet)
 Apr. 19: Revised due date, project 2
Week #14:
 Apr. 22:
 Finish this morning's Chisquared problem: In a
(fictional) survey of 145 randomly chosen WU students living in the
dorms, we get this data:

Dorm\Meal Plan

A

B

C

D

East side

9

21

21

11

West side

8

14

13

6

Kaneko & others

6

13

15

8

So answer whether dorm & meal plan are independent. Note the
P value is instructive here, so find a specific value if reasonably
possible (maybe an online Chisquared calculator if not your own
calculator).  Do 13.67, 13.70,13.71,13.73, and ... 13.75, and 13.77 (there is a red flag in the data. What is it? If we ignore it, despite the text's warning, we get the answer in the back, so go ahead for practice.)
 Apr. 24:
 Read the ANOVA handout, and start to set up problem 15.17 (p. 513) using ANOVA
 Do 13.80, 13.81 (for 13.81, review Poisson to find the
probabilities for each count, and hence the expected number, out of 300
measurements, of each number of counts, then finish)
 do 13.83b,c
 Apr. 26:
Week #15:
 Apr. 29 (Last day of class)
 Do/Finish 15.17 (p. 513). I don't recall if we should reject H_{o} or not in this problem, but if we do, what does it mean, specifically? (What does the ANOVA tell us, in practical terms?)
 A regression problem to look at:
 Data
set: (7,7),(9,11),(11,15),(13,9),(15,13),(17,19),(19,17)
[Note: Data chosen to come out nice; use the verbal procedure we
developed in class.]
 Find the correlation coefficient r.
 According to r, how strong does the correlation appear?
 Plot the points to see how close to a line the points actually lie.
 Figure out how to get your calculator to find r for you.
 Use r to find the regression line coefficients.
 What are the "predicted" yvalues (the "yhats") for each x in the list {7,9,11,13,15,17,19} ? What is the sum of squares of the errors (yy_{hat})?
 Some Final Exam Study Aids:
 An old final exam from a
related class. PLEASE NOTE that this exam has one (or perhaps
two) problems I wouldn't quite ask you in this class, and it's much
longer than what I'd likely give you. I'm sure there is more than
one topic we've covered this term that we didn't cover in the other class. All that said, these are mostly good problems for you to be able to do.
 The Survival Guide
is still appropriate reading. Do some serious organization to
decide what topics you need to study. You might start by asking
yourself what topics we have studied do not appear in the sample final above.
 Remember, Q&A session on Thursday, May 2 at 7 p.m. in Ford 201. Bring questions, as I will not bring a prepared program.
 Apr. 30  a little bonus Bayes' Theorem review problem to puzzle over while
you study for your final exam (brazenly stolen from the World Wide Web):
Two balls, B_{1} and B_{2},
are placed in an urn. Each ball has equal probability of being
black or white
(alternately, we toss a coin to choose which color to
put in for ball 1, again for ball 2). Someone reaches into the
urn without looking and draws out a black ball.
 What is the chance that the urn contained two black balls?
 Having trouble understanding the question? Imagine we
repeatedly do the whole business of setting up the urn and drawing one
ball. Whenever the draw is black, you declare "the urn is all
black!", but if the draw is white, you say nothing. In the long
run, what percentage of the time were you correct?
 Once you have the first answer, this one is harder: We
set up the urn as above, and the draw is black. Putting the drawn
ball back in the urn, what's the chance that a second draw is black?
Week #15/16:
 Saturday, May 4, 811 a.m.. Final Exam
Last modified April 30, 2013.
Prof.
Janeba's Home Page  Send comments or
questions to:
mjanebawillamette.edu
Department of Mathematics  Willamette University
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