### 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

### 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 (six-sided, "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 non-green, i.e. blue and orange are treated as the same result - non-green)?  How about exactly 2 greens?
• Also, do p. 15 #1.1, 1.2, 1.5, 1.6, 1.9, and p. 19 #1.27 .
• 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:
• 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
• Do 2.17, 2.21, 2.23, 2.76,2.79,2.89, 2.93
• Jan. 25:
• Do these problems (the questions are long, but the answers aren't proportionally long).
1. 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(B|A) and P(B|A') 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.
2. 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 non-users, i.e. reports 99% of them as non-users, 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(B|A) and P(B|A') mean in practical terms for the athletes - give the numbers AND explain what those numbers are telling us.
• Now explain what P(A|B) 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.
3. Do text problems 2.95, 2.97 (you may assume the town is so large that non-replacement does not significantly change the odds of selecting someone with a deficiency), 2.99, 2.109

### Week #3:

• Jan. 28:
• Do these problems:
1. (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, six-sided die.
2. Now suppose you have a 12-sided 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.
3. 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.
4. One unusual way to define the constant e is that e is the limit, as n goes to infinity, of (1-1/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 one-car 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 non-obvious 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 case-sensitive 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.
• Re-read 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 65-year-old 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).
• 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 6-8 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:

• 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. {1,2,3,4,5,6,7}
2. {0,0,0,2,2,2}
3. {5,5,5,5}
4. {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).
• 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/(1-a) 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(y-x) 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 to compute and verifying you get back to where we started.
• Do 3.54, 3.55, 3.58,

### Week #7:

• Feb. 25:
• 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 5.1-5.4 (5.1-5.3 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 5.5-5.6; 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 one-sentence non-technical 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 extra-credit opportunity:  Today in class we derived the moment-generating function for the standard normal distribution; it was MX(t) = e/2Expand 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/(2²).  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 = (W-µ)/, 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
• 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 oft-repeated rule of thumb, that if n>30, then Xbar's distribution may be approximated as normal, although if X's distribution is already normal, Xbar 's will be as well, for any value of n.
Suppose that X1,X2, and X3, have a uniform distribution on [0,1] (see section 6.2).  Thus each Xi takes on values between 0 and 1.
1. What is a possible range of values for Xbar in this context?
2. Using the information about the uniform distribution in 6.2 and what we read about Xbar in 8.2, show that the criterion in Prof. Janeba's rule of thumb, "Xbar'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 18-24 in the US have average height 70" with SD 2.5" (the values are approximate, assume for this problem they are exact).
1. Then for a random sample of 400 of these males, the average will be _____ (mean for Xbar) give or take ______ (SD for Xbar) or so.
2. 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).
1. The proportion in the sample that are female will be _____ (mean for Capital Thetahat), give or take ______ (SD for Capital Thetahat) or so.
2. What is the chance that the sample proportion will be under 50%?
• 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 referred-to 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.
• Apr. 3: Quiz 8; Project 2 assigned
• Catch up on confidence interval homework & work on project
• A short-essay 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:
• 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 2-3 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:
• Re-read 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 hypothesis-test 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 Ho - a large P value or a small one?  Alternately,
• If we agree to test at the =0.05 level, then we reject Ho if P is (pick one: larger, smaller) than 0.05.
• If we reject Ho 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 Ho is true.  Ho 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 fi,j and ei,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 Chi-squared 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 Chi-squared 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 Ho 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" y-values (the "y-hats") for each x in the list {7,9,11,13,15,17,19} ?  What is the sum of squares of the errors (y-yhat)?
• 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, B1 and B2, 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, 8-11 a.m..  Final Exam