Math 308/361: A Sampler of Problems
http://www.math.uiuc.edu/~hildebr/361/sampler.html
Here is a preview of some of the problems that can be treated with
the methods developed in this course. Some of the problems will
be discussed in class, while others may show up as homework problems.
The goal of the course is to develop tools to
solve problems of this sort theoretically.
Many of these problems lend themselves to computer simulations,
and is of interest to come up with (approximate) answers via
such simulations and compare these answers with those derived by
theoretical methods.
In some cases ready-made java applets are available
to carry out these simulations online; I have provided links
to such applets. If you are curious, try
one of these simulations, or write your own programs.
Simulating the Florida election
In the Florida election, out of some 6 million votes, the number
of votes for Bush and Gore were within a few thousand of each
other.
This raises the following interesting probability question: If each of
the 6 million votes had been cast by tossing a coin, would it be
reasonable to expect the number of heads and tails to differ by
no more than a few thousand? For example,
what is the
probability that in 6 million coin tosses the number of heads and
tails differ by at most 3,000? What are the corresponding
probabilities if 3,000 is replaced by 300, or by 30? What about the
extreme case, when the difference between heads and tails is zero?
In other words, what is the probability
that out of 6,000,000 coin tosses exactly half (i.e., 3,000,000) end
up heads?
Computer simulation: It is very easy to write a program to
simulate repeated coin tosses. If you are so inclined, do that to
get a feel of what to expect.
Polling
Continuing with the election theme, here is
problem related to polling:
A polling firm wants
to determine the proportion of likely voters
who favor the incumbent candidate in an upcoming election. How many
voters does it have to poll so that, with 90 % probability, the polling
error is at most 2 percentage points?
De Mere's paradox
Which is more likely, getting at
least one six in 4 rolls of
a die (Game I),
or getting at least one double six in 24 rolls of 2 dice (Game II)?
The Chevalier de Mere (1610--1685) argued (incorrectly, as it
turned out) that
both probabilities
are the same (and equal to 2/3), since 4 times 1/6 is 2/3
and 24 times 1/36 is also 2/3. What are the correct probabilities
for winning in these two games? Which (if any) of the two games
gives you a better than 50-50 chance of winning?
Computer simulations:
Java applets with computer simulations of both games are available,
for both games,
Game
I, and
Game II
(courtesy of the Chance
Website)
Try to determine the probabilities in question
experimentally by playing these games a sufficient number of times,
and try to determine which of the two probabilities (if any) is
greater.
Alternatively, write your own program to simulate the two games.
The Birthday Paradox
A rather well-known (apparent) paradox is that a relatively small
group of people is sufficient to ensure, with very high
probability, that there are at least two members of the group who
have the same birthday. For example, it takes only 23 people to have
a better than 50-50 chance of finding a matching birthday.
A java applet with nice graphics simulating this birthday paradox
is available
here. (This is part of the
Probability by Surprise collection of java applets.)
A different java applet, which computes the probabilities for finding a
matching birthday in a group of n people, can be found
here (again from the Chance Website).
Related to that paradox is the following problem, suggested by a reader
question in the ``Ask Marilyn'' column in
Parade Magazine, January 11, 1998. [From the
"Chance News" section of the Chance Website; "Chance News" is a monthly
compilation of newspaper and magazine articles that raise
interesting probabilistic or statistical questions.
I started counting from the first person who came
came into the office. I counted until I found a
matching birthday in the group. Then I started a new
survey with the next person. In eight surveys, the
smallest number of people it took before I found
a matching birthday was 12. The largest number was
only 54.
Why is a minimum of 12 not surprising?
Specifically, what is the probability that
there is no match in the first 12 people in 8 surveys?
Baseball World Series
In the baseball world series, two teams play against each
other. The first team to win 4 games is declared the winner. Thus,
a world series lasts at most 7 games and at least 4 games.
Suppose A is the better team and has a 2/3 chance of winning a game
against B. What is the probability that the series is
over in 6 or fewer games?
Airline overbooking
Suppose an airline accepted 12 reservations for a small
commuter plane with 10 seats. They know that 7 reservations went to
regular commuters who will show up for sure. The other 5 passengers will
show up with a 50 % chance, independently of each other. (a) Find the
probability that the flight will be overbooked; (b) Find the
probability that there will be empty seats; (c)
Find the average number of passengers turned away.
Records
A die is rolled repeatedly. Saying that a record
occurs at roll n means that the number appearing on the n-th roll
is strictly greater than the number appearing on all previous rolls.
The number appearing on the first roll is, by default, considered to be
a record.
(a) What is the (numerical) probability that a record occurs at the
4th roll?
(b) Find a general formula (which may involve a summation)
for the probability that a record occurs at
the n-th roll, for n=1,2,...
Math 361 Course Web Page
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Last modified Tue 21 Jan 2003 09:11:16 AM CST