Discussion Sessions and Supplementary Materials for
Below are some "Discussion Sessions" to be used with students in conjunction
with the book Struck by Lightning: The Curious
World of Probabilities. Each Session consists of a short reading
assignment from the book, together with some questions to inspire
discussion. The intention is that the students will first do the reading
and make notes about the questions on their own (e.g. as homework), and
then the entire class will have a full discussion of the issues raised.
(The students should also be invited to offer additional observations /
comments / questions / opinions about the readings.)
I have found that the Sessions lead to class discussions usually lasting
approximately 45 minutes.
The book and questions are appropriate for group study or for highschool
or university students. You are welcome to use these Discussion Sessions
in your own teaching. If you do, I would appreciate your letting me know.
Below are also links to some "Supplementary Materials" (in
pdf format) for inclass smallgroup exercises. They are listed
under the chapter to which they most closely correspond, but many of them
can be used at virtually any point in the course. Their levels and
lengths vary, but generally they are slightly more
mathematical/technical exercises which expand upon the book's material.
They usually take about 4560 minutes
including followup discussion, and work well for groups of 34 students
with limited mathematics background.
Discussion Session #1: Introduction
Read Chapter 1 (pages 16), and then answer the following questions.

Make a brief pointform list of the 34
different kinds of randomness, mentioned in the chapter, that you found
the most interesting (and why).

List 23 kinds of randomness (whether mentioned in the chapter or not)
that have affected your life in some interesting way (and explain how).
Supplementary Materials:
intro,
gambler
Discussion Session #2: Coincidences, Part I
Read the beginning of Chapter 2, from page 7 to the end of
page 14. While you read, consider (and make notes about) the following
questions:

What does the book mean by "out of how many"?

Consider the following six "stories" from the reading:
the lottery winner, the ten coin flips,
Disney World, the friend's dream, Richard Feynman, the molecules of
water, and Darth Vader versus Lord Dark Helmet.
For each of these six stories:
(a) Provide a onesentence summary of the story.
(b) Explain (in one more sentence) what we can conclude from the story.

Describe Milgram's "six degrees of separation" experiment.
Had you heard of it before?
Do you find it interesting?
What can we learn from it?
What flaws did it have?

What are "Erdos numbers"?
What other similar "numbers" have been developed?
Do you find them interesting?
Can you suggest any other similar "numbers" not mentioned in the book?

Think of two or three "coincidences" that have occurred in your own
life. For each one, answer the following questions:
(a) Did you find it surprising? Why or why not?
(b) Did you feel that it happened just by chance, or that it
had some special significance? Why?
(c) To what extent do you feel that it can or cannot be
explained by considering "out of how many"?
Supplementary Materials:
seating
Discussion Session #3: Coincidences, Part II
Read the end of Chapter 2, from page 15 to the end of
page 22. While you read, consider (and make brief
notes about) the following questions:

Why is it so much more likely that some pair of people at a party
will have the same birthday, than that someone at the party will have
their birthday today? Explain.

Do you think that some pair of people in this class has the same
birthday?
Why or why not? What if we also include everyone's mother's
birthday, as well?

Why does the value 0.69 (on page 16) represent "a little bit of
overcounting"?

How do you think we could calculate the exact probability that
some pair out of 23 people has the same birthday? [Hint: This is equal
to one minus the probability that no pair has the same birthday,
i.e. that every birthday is different.]

What is the conclusion of the "Musical Mayhem" story? Do you find it
surprising?

What is "Poisson clumping"? How is it related to Figure 2.1? How is
it related to the five Toronto homicides in the first week of November,
2003?
What other examples of Poisson clumping does the book allude to?

What is the "common cause" explanation? How does it relate to the
"Parental Musings" story?

Think of at least two examples of coincidences from your own life, or
that you have heard or read about, which you think can be explained by
"number of pairs" or "Poisson clumping" or "common cause".
Describe your reasoning.
Supplementary Materials:
fives,
matchings
Discussion Session #4: Law of Large Numbers, Part I
Read the beginning of Chapter 3, from page 23 to page 30 (stop at
"Another game is keno"). While you read, consider (and make brief
notes about) the following questions:

What are the "two key facts" which explain why casinos always make
money?
Did you already know these facts? Do you find them surprising?

What is the Law of Large Numbers, and how is it related to casino
profits?

What can we conclude from the "Honorable Bus Fine" story?

What are all the different roulette bets which are considered?

For each of those bets, how is the average outcome calculated?
What does it work out to be?

What "gambler's ruin" question is considered (p. 27)?
What answer is provided? How does this
answer compare with what you would have guessed?
How difficult do you think it would be to compute
this answer mathematically?

What can we conclude from the "Slowly but Surely" story?
Supplementary Materials:
gambler1 (longer; could save until later),
mathrem,
gambler2
Discussion Session #5: Law of Large Numbers, Part II
Read the end of Chapter 3, from page 32 ("Rolling the Dice")
to the end of page 43. While you read, consider (and make brief notes
about) the following questions:

Summarise the three different explanations for why, when rolling two
ordinary dice, the most likely sum is 7. Did you already know that
fact? Do you find the explanations convincing?

Explain how to compute the probability of getting at least one 3, if
you roll an ordinary die 4 times. And of getting at least one pair of
6's, if you roll a pair of ordinary dice 24 times. What is the
historical significance of these calculations?

Explain the rules of craps, and the probability of winning.

Do you know (or can you easily figure out)
how to compute the probability of winning at craps? If yes,
explain how. Or, if no, explain why it is difficult, and whether or not
you would like to learn how.

What is a "Don't Pass Line" bet? Why does it not give the edge to the
player?

Summarise the examples presented in the section "A Life of Large
Numbers". Do you find the asserted connections to the Law of
Large Numbers to be convincing? Why or why not?

Come up with at least two examples from your own life, of events which
are somehow related to the Law of Large Numbers.
Supplementary Materials:
craps
Discussion Session #6: Card Games and Beyond
Read the beginning of Chapter 4, from page 44 to the end of page
51, and also from the middle of page 60 ("Patience, Patience") to the
end of page 62. While you read, consider (and make brief notes about)
the following questions:

What is the point of the Larry Bird and bowling examples?

Do you play bridge? Poker? Other games of chance? Do you enjoy them?

In bridge, what is a "finesse"? How do expert players differ from
novices when it comes to finesses?

What is duplicate bridge? To what extent does duplicate bridge
eliminate luck from bridge? Explain.
How is this related to the "Bridge Bickering" story?

In poker, what is the probability of successfully drawing to a Flush?
How does this depend on the opponents' faceup cards?

What is the difference between drawing to an Inside Straight and an
Outside Straight? How are the probabilities related? Why?

Explain the probability calculation in the "Showdown" story on page 50.

What are "pot odds"? How can they help make decisions when playing
poker? Explain the pot odds calculation on page 51.

Why is "patience" required to be successful at games of chance? How
is this related to the Portuguese Postal Puzzle story?

Think of at least two examples from your own life in which
you had a high probability
of winning or succeeding at something, but you still didn't win due to
bad luck and insufficient repetition.
Supplementary Materials:
smallsum,
pokerprobs,
flush
Discussion Session #7: Homicide Counts and Rates
Read the beginning of Chapter 5, from page 63 to the middle of page 68
("... how large is the decline?"). While you read, consider (and make
brief notes about) the following questions:

According to the book, what segments of society declare that violent
crime is increasing? What evidence does the book give for this?

Can you think of any other examples of people declaring that violent
crime is increasing?

What is the difference between homicide counts, and homicide rates?
Why is this distinction important?

Give at least four examples from the book
of pairs of countries or regions, for which
one of them has a higher homicide count, and the other one has a higher
homicide rate.

What is the connection between homicide rates versus counts, and
the "out of how many" principle from Chapter 2?

Describe the data in Figure 5.2, noting as many features as you can.
What do you think we can and cannot conclude from this graph?
Explain your reasoning.

In the summer of 2005 (just after the book was written),
the city of Toronto experienced a higherthanusual number of
homicides involving handguns, leading to many alarming headlines
("summer of the gun", "Toronto has lost its innocence",
"guns used to bathe Toronto in blood", ...).
Do you feel that this was the start of a new
and dangerous period of violent crime, or merely a statistical
fluctuation that will soon disappear? Explain your reasoning.
The following data may help:
Homicides per 100,000 population of various metropolitan regions in
2005:
Toronto 1.96;
Winnipeg 3.72;
Regina 3.97;
Edmonton 4.29;
New York 6.60;
Los Angeles 12.6;
Chicago 15.6;
Buffalo 19.8;
Detroit 39.5.
All of Canada: 2.04.
(Toronto in 1991: 2.41.)
Discussion Session #8: Homicide Trends
Read the end of Chapter 5, from the middle of page 68 ("Measuring Trends:
Regression") to the end of page 77. While you read, consider
(and make brief notes about) the following questions:

What is "regression", and how does it avoid bias and subjectivity?
How does it relate to the "Weight Gain, Weight Loss" story?

Explain the purpose and conclusion of each of the first five figures
of Chapter 5, i.e. Figures 5.1, 5.2, 5.3, 5.4, and 5.5. Also explain
the relation between the different figures: do they present
different information? are they all necessary? why or why not?

Summarise the comparison of homicides in the U.S., Canada, Australia,
and Great Britain. Do you find any of the comparisons surprising? Why
or why not?

Summarise the content of Table 5.3.
Do you find any of the information surprising? Why or why not?

Summarise the comments about crime rate trends in the section "Playing
with Numbers?" What is your reaction to these comments?
Why does the section conclude by saying "Just the facts, ma'am"?

Last Session, we discussed the question of whether Toronto's
higherthanusual number of homicides in 2005 represented the start of
a new and dangerous period of violent crime, or merely a statistical
fluctuation that will soon disappear. Has your opinion on this question
been affected by anything from this Session? Why or why not?
Discussion Session #9: Utility Functions
Read the book from the top of page 87
to the end of page 95.
While you read, consider (and make brief notes about) the following
questions:

Summarise the two decisions made in the story "Walk or Ride?" Do you
agree with the decisions? Why or why not?

What are utility functions?

How were utility functions used in the book
to decide about the wedding, and about phoning "Juan"?
(Explain the calculations used.)

Do you personally believe that expected values of
utility functions are a good way to make decisions? Do you think that
most people make decisions this way (whether they realise it or not)?
Why or why not?

Summarise the calculations in the book
about the insurance policy. Why does the
book say that insurance can sometimes be a winwin situation?
Do you personally believe in purchasing insurance? Why or why not?

Summarise the various examples of differing utility functions in the
section "Your Utility or Mine?" Give an example from your
own life where two people had significantly different utility functions,
and a conflict that arose as a result.

On a scale where seeing a pretty good movie equals +10,
decide (with brief explanation) on the value of your own personal
utility function for each of the following events:
(a) getting caught in the rain without a jacket;
(b) finding a $20 bill; (c) losing a $20 bill; (d) winning one million
dollars; (e) winning ten million dollars.

Based solely on your utility function values, would you be
willing to run through the rain without a jacket
(a) in order to pick up a $20 bill you found?
(b) in order to pick up a
lottery ticket that had one chance in a million of being worth ten
million dollars?
(Explain your calculations.)

Think of an example (as different as possible from those already
mentioned in the book or the previous question) of an event which is
quite unlikely, but which would have significant consequences. Give a
rough guess of its probability, and also decide (with brief explanation)
the value of your utility function for the consequences
(again, on a scale where seeing a pretty good movie equals +10).
Supplementary Materials:
utility,
utility2
Discussion Session #10: Medical Studies, Part I
Read the book from page 96 to the middle of page 106 (ending with
"... publicize their product"). While you read, consider (and make
brief notes about) the following questions:

Summarise the "Lucky Shot?" story.
What answer would you give to the boyfriend at the end?

What does it mean to "just get lucky"?

What are pvalues? (Give as much detail as possible.)

What is the "5% standard"?
Do you feel that 5% is an appropriate standard? Why or why not?
Does it depend what is being studied?

Discuss the various calculations related to
the Probabilitus example and the candies example.

In the Probabilitus example, suppose instead that 70 patients out of 100
were cured. Do you think this would this "prove" that the drug works?
Why or why not? What mathematical quantity would be important for
determining this?

What is the principle of "regression to the mean", why does it
arise, and what examples does the book provide?

Explain "sampling bias" and "reporting bias". Summarise the
book's examples of them. Think of two additional examples not
from the book.

On July 31, 2005, the Globe and Mail newspaper's web site
conducted an online survey, in which they asked,
"Which medium do you rely on most to keep abreast of the news?" They
received 1623 responses, and the results were:
Television, 194 votes (12%);
Radio, 110 votes (7%);
Newspapers, 298 votes (18%);
Internet, 1013 votes (62%);
Magazines, 8 votes (0.5%).
To what extent does this survey "prove" that most Canadians receive most
of their news from the Internet? Explain. How is this related to
medical studies?
Supplementary Materials:
medical,
medical2,
regtomean
Discussion Session #11: Medical Studies, Part II
Read from the middle of page 106 (beginning with "Publication Bias")
to the bottom of page 114 of the book. While you are reading,
consider (and make notes about) the following questions.

What is "publication bias"?

Summarise the stories about each of: asking your mom/dad;
Happiness Hats; Dr. Nancy Olivieri; Dr. Betty Dong; and Vioxx. What do
these stories all have in common? What can we learn from them?

What did the International Committee of Medical Journal Editors declare
in 2001? How is it related to publication bias? Do you feel that it
will completely solve the problem of bias in medical studies?
Why or why not?

What does it mean that "correlation does not imply causation"?

Summarise the stories about each of: the Jumping Frog; cigarettes and
yellow stains; the Meditation Medical Miracle; the medical school class
presidents; and television versus violence.
What do these stories all have in common? What can we learn from them?

What are "randomized trials"? How do they relate to correlation
and causation? Do you think they solve all problems of interpreting
causation? Why or why not?

The next time your doctor tells you to take a certain drug or action
because studies prove it is beneficial, how will you react? Why?
Supplementary Materials:
medical3
Optional MidCourse Amusement:
Read the book's "Interlude" (Chapter 9, pages 133144) for a little
fun reading. Note any comments or questions or reactions that you have.
Then, if you wish, make up your own humorous probability story  just
a few paragraphs long  in the spirit of the book's Interlude.
(You can later read your story out to the class.)
Discussion Session #12: Election Polls
Read the beginning of Chapter 10 of the book, from page 145 to the
end of page 153. While you are reading, consider (and make notes about)
the following questions.

What elections (including referenda) are referred to in these readings?
(List as many as you can.) Which of them had you already heard of?

Why does the book claim that "polls provide us with more direct
democracy than elections do"? To what extent do you agree or disagree?
Why?

Why does the book claim that polls "provide an extra level of
communication among the electorate"? What examples are given to support
this claim? To what extent do you agree or disagree with it? Why?

According to the book, what does a poll's "margin of error"
really mean? What factors does it take into account? What factors does
it not take into account?

What examples does the book give of elections (and referenda) in
which voter opinion changed after polls were conducted? (List as many
as you can, and provide as much detail as possible.)

What examples does the book give of situations in which respondents
will provide dishonest or misleading answers to pollsters?

Summarise the (fictional) story about the "Apathy Party", and explain
what we can learn from it.

Explain the example of the cannabis survey, and how a "randomized
response" poll could help.

Given everything you have learned so far about election polls,
to what extent do you think they can or cannot
be used to predict the results of an upcoming election? Explain.
Supplementary Materials:
pollster
Discussion Session #13: Margins of Error
Read pages 164 to 171 (inclusive) of the book, and also the story
"An Uncertain Day" on pages 174175. While you are reading, consider
(and make notes about) the following questions.
 From a probability perspective, in what way is conducting a poll
like flipping coins? In what way is it different?
 What does the book claim is the "95% Range for Percentage of
Heads when Flipping 10 Coins"? What does this mean?
 What about for 100 coins?
 What formula does the book claim for the margin of error when
flipping lots of coins? Do you understand how it was derived?
How is it related to the "bell curve"?
How is it related to the margin of error for polls?
 For each of the four polls summarised on pages 145146:
(a) How closely does the published margin of error follow the claimed
formula?
(b) How close was the poll's result to the actual election result?
(c) Was the poll's true error (compared to the actual election result)
more or less than the published margin of error?
 Find some election poll not mentioned in the book (e.g. from the
internet, or from a current election), which includes the number of
people sampled and a margin of error. How closely does the
published margin of error match the book's formula?
 What examples does the "Uncertain Day" story give about how
margins of error might arise in everyday life? For each example,
do you agree that it is a valid illustration of margins of error in
everyday life? Why or why not?
 Present at least two additional examples, not mentioned in the
book, of how margins of error might arise in everyday life.
Supplementary Materials:
buffon
Discussion Session #14: Randomness to the Rescue
Read Chapter 12 from page 176 to the middle of page 185 (ending with
"but to create it"), and also all of page 192. While you are reading,
consider (and make notes about) the following questions.

What action does the book claim gives results that the CIA couldn't
reproduce in a million years? Do you believe this? Do you find it
surprising? Why or why not? Does it give you any new perspective about
your daily activities, and if so, what?
 Give an additional example (not from the book) of
types of things people frequently do, which have probably never occurred
before precisely that way in all of human history.

Explain the stories of the million monkeys and the Internet
searches. Had you heard of them before? Do you find them surprising?
Why or why not?

Perform a few Internet searches of your own (on search
strings not from the book), using the Google search engine, keeping track
of the number of hits each time, to further illustrate the book's
story about Internet searches. You could try random digits, random
letters, or even random sequences of words (within quotation marks).

Explain the examples about Johnny Hooker, Rock/Paper/Scissors, and the
World Series.
Did you find them convincing?
What do they all have in common?

Do you think that the game Rock/Paper/Scissors involves some skill, or
is just pure luck? After considering this question, try out the NYT
RPS robot (on "Veteran" mode); how did you do, and how does that
affect your answer to this question?

Summarise the uses of randomness by computers. Do you find any of them
surprising?

What are "pseudorandom numbers"?

Summarise the story "Dividing the Restaurant Bill". Do you feel that
the solution reached was fair? Why or why not? How is the solution
related to the Law of Large Numbers? How are these questions related to
this
auction house newspaper story?

Have you ever considered using such a method in the past?
Would you consider using such a method in the future?
Supplementary Materials:
rng
Discussion Session #15: Randomness in Biology
Read the book from the middle of page 195 (beginning with "Designer
Blue Genes") to the end of page 205.
While you are reading, consider
(and make notes about) the following questions.

What did you previously know about the importance of probability in
biology?

What rule does the book give for genes being passed from parent to
child? How is this related to eye colour?

What does the book claim about children of two Lighteyed parents?
Does this claim agree with your observations about yourself and/or your
family and friends?

What does the book claim about children
of one Lighteyed and one Darkeyed parent?
How about two Darkeyed parents?

What does the book claim about the eyecolour genes of the
book's author?

How is the spread of viruses related to probability?

Summarise the story "Spread the Word". How is it related to the
spread of viruses?

What is "herd immunity"? How is it related to HIV/AIDS and to
vaccines?

Summarise the history of the MMR vaccine in the
United Kingdom. What can we learn from this story? (See also this
update.)

What does the book say is the "total fertility rate" of the world?
Of Canada? What is the significance of these figures?

Recently, Canada has had yearly totals of about 340,000 births,
230,000 deaths, 240,000 immigrants, and 40,000 emigrants. What does
this say about Canada's population change in recent years? How is this
related to the previous question?

What does the book mean by "selfreplicating"? What
selfreplicating examples are given? What to they all have in common,
related to probabilities?

What other sciences (besides biology) do you think are related to
probability? In what ways?
Supplementary Materials:
genetics,
disease
Discussion Session #16: Conditional Probability and Spam
Read the book from the top of page 206 to the middle of
page 209 (ending with "... for any other contestant"); and also
all of page 220; and also from the top of page 224 (beginning with
"Block That Spam") to the middle of page 229 (ending with "...
new spamfiltering computer programs").
While you are reading, consider
(and make notes about) the following questions.

Summarise the (brief) stories about crossing the road, passing an
examination, and "the woman of my dreams". What do they have in
common?

Summarise the stories about each of
the bearded men, the lupus scare, and the spelling bee. (Include
details of the calculations made.) What can we learn from them?
 What common societal situations/reactions are analogous to this
"Beware the Bearded Men" story? (Think of as many as you can.)

What is spam email?

Why does spam email have that name?
[See also
the original
Monty Python spam skit.]

What is automatic spam filtering?

What does the book describe as a "first attempt" at automatic spam
filtering? How does it work? What are its limitations?

Summarise the stories "The Dastardly Spammity Spam" and
"Spammity Spam Revisited". What can we learn from them?

Explain probabilitybased spam filters
(giving as much detail as possible).
Do you think they are more effective than
previous spam filters? Why or why not?

Read the blog post
"Adventures in
spam, part two".
How is it related to the above issues?

Does your own personal email account have a spam filter
(e.g., a "junk folder")?
Approximately how many spam messages per week does it catch?
How many does it miss?
Do the spam messages
ever contain intentionally misspelled words like "Vi@gra"?
Supplementary Materials:
conditional
Discussion Session #17: Causes of Randomness
Read all of Chapter 16 (pages 234 to 246). While you are reading,
consider (and make notes about) the following questions.

What does the book mean when it says that randomness is often
based on ignorance?

Summarise the story of the Restless Restauranteur, and relate it to the
previous question.

How does the book explain the randomness of flipping a coin?

What does it mean for a system to be chaotic? (Give as
much detail as possible.)

What examples of chaotic systems does the book give?
(List as many as you can, and discuss whether or not you agree that
they are chaotic.)

What are some other examples, not mentioned in the book,
of chaotic systems? Of nonchaotic systems?
(Try to think of at least two of each.)

Do you agree with the book's assertion
that "meteorologists are ambassadors of probability"?
Why or why not?

Do you often listen to weather forecasts? If yes, then how do
you feel about them? Do you think they are usually accurate? Do they
describe probabilities in a way that you find clear?

Are you surprised that probability theory is related to physics? Why or
why not?

What does the theory of quantum mechanics say about nature and randomness?
(Give as much detail as you can.)

How are chaos and quantum mechanics related to the random numbers used
by computers? (Provide as many connections as you can.)

Do you believe that nature is truly random? Why or why not?
If yes, then does this affect your perception
and feelings about the universe, and if so how?
Optional EndofCourse Amusement:
Read aloud the book's mock "Final Exam" on pages
247254, taking turns reading each question (including all
the multiplechoice answers).
Then, as a group, come up with at least three additional "Final Exam"
style questions about this course, in a similar style to those in the
book (i.e. multiple choice, obvious answer, humorous, etc.).
(You will later read some of your questions out loud to the entire class.)
The book Struck by Lightning,
by Jeffrey S. Rosenthal,
may be ordered from e.g. amazon.ca
(or in paperback)
or indigo.ca (or in paperback) or amazon.com or amazon.co.uk or barnesandnoble.com, or from many bookstores.
See also the
Struck by Lightning main page.