Essay Topics
Your assignment is to write a short essay/solution (about 2 pages) on
one of the topics below. You may choose a topic not on the list if you wish,
but you should run it by me first. Most of these questions are
"open ended", and are designed to get you thinking about some things
that we didn't cover in detail in class. Look
here for instructions on the format of your essay and a description
of what I'll be looking for.
Topics
You did not need to limit your essay to the questions given. Feel free
to ask and answer your own questions as well.
Coincidence
The Telegraph-Tribune, 8 Nov. 1996, B2, Jeff Ballinger:
``Last Fall Johnathan Wince was a freshman in high school in Paso Robles
California and his brother Christopher was a senior. While sitting in
a ceramics class with Christopher, Johnathan signed and dated several dollar
bills and these bills were later spent by the brothers.
``In September, of of this year, Christopher was a freshman at Morris
Brown University in Atlanta. To his great surprise, Christopher got
back one of the dollars in change while shopping in Atlanta's underground
mall."
Questions:
A reporter asked a statistician "What is the chance of that happening?",
but the paper did not print his answer. How would you answer?
Give the assumptions you are making, and discuss how likely they are
to be true.
Weather Prediction
The six o'clock weather predicts the weather. How would you assess their
accuracy level?
Station A either predicts "rain" or "no rain" for tomorrow. Station
B provides a probability of rain. How would you compare the accuracy
of prediction of these two stations?
You can learn more about assessing weather prediction in Statistical
Methods in the Atmospheric Sciences by Daniel S. Wilks, Academic
Press, 1995.
Random Numbers.
Ask Marilyn.
Parade Magazine, 29 Oct. 1995, p. 18
Marilyn vos Savant
A reader writes:
From what I understand, computers do exactly
what you instruct them to do. If so, how can
a computer generate random numbers? How does
someone write one set of instructions that would
produce a different answer every time? It seems
like an impossible task.
Michael H.G. Ho,
Sagamore Hills, Ohio
Marilyn answers:
Computer programs don't generate numbers randomly,
but the results can be used as random numbers.
That's because they're likely to be "more random"
than any other random number generator you can name,
such as a deck of cards, or the fellow at the desk
next to yours. After all, how can anything be truly
random?
How would you answer the reader's question.
Then, how can anything be truly random?
An entertaining discussion on how to determine if a sequence of numbers is random can be found in The Broken Dice, Ivar Ekeland, University
of Chicago Press, 1991, Chapter 1.
Number 6: DNA Fingerprinting
DNA Fingerprinting was controversial long before the OJ Simpson
trial. Read the article "Odds you just can't grasp" from
the NY Times, dated 12/19/94. The article touches on a few of the
arguments surrounding the use of so-called DNA "fingerprinting" in trials.
Notice the cameo appearance by Bruce Weir, who later went on to testify
for the prosecution
at the Simpson trial and embarrassed himself by making some arithmetical
mistakes.
You may want to address these questions:
- What, in your own words, is the prosecutor's fallacy?
- The article provides an explanation of the prosecutor's fallacy.
If you were a prosecutor, and the defense used such an explanation, how
might you respond?
- How would you interpret the statement "'the
defendant has the same DNA profile as that left at the crime scene by the
perpetrator, and this type is estimated to occur in one in a million people"
in the context of a criminal trial?
- Some people have criticized prosecutors who quote extremely small
probabilities for matching the defendent to the DNA at the crimescene.
For example, if I say that the probability that such a match occurred
by chance alone is 1 in 6 billion, what does this mean, given that there
are only 5 billion people in the world? What do you think this means?
Does this bother you? Why or why not?
- Can you think of any other examples of probabilities that are
often misunderstood by the public?
- What probability concepts have you found to be counterintuitive?
Monty Hall
This problem has a right answer and a wrong answer; make sure you get
it right!
From Marilyn Vos Savant, "smartest woman in the world" and columnist
at Parade Magazine:
"Suppose you're on a game show, and you're given the choice of three doors:
Behind one door is a car; behind the others, goats. You pick a door, say No. 1,
and the host, who knows what's behind the other doors, opens another door, say
No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?'
Is it to your advantage to take the switch?"
What do you think? Why? What assumptions are you making? (Hint: you
should probably assume that the host knows where the car is.)
This problem caused quite a controversy when Ms. Vos Savant published her
answer. You might want to look into this when you write your essay.
It's rather unintuitive, and some people refuse to believe in the answer.
Newspapers
Newspaper reporters often get statistical and probabilistic concepts wrong,
or they oversimplify so that a statement is vacuous or incorrect. Find
an example of a statistical or probabilistic argument or discussion in
the newspaper. Describe the point the article is making (or the people
in the article), and criticize it. Do you agree or disagree? Are they
correct or incorrect? The New York Times science section is a great
source. (It's on-line, of course.)
Be sure to include a copy of the article.
Home
Instructions on writing the essay.