illustrate the technique of the Monte Carlo simulation approach to
estimate probabilities for two geometrically oriented problems involving line
segments and another circles within the unit circle.
Algebra, precalculus, discrete math, courses involving
elementary probability, or introductory programming courses.
Familiarity circles, radius, distance between points, and midpoint of a line
segment; elementary probability ideas; if used in a programming course, commands
for random numbers, square roots, conditional statements, and
loops; if graphics are included, parametric representation of a circle.
The animations included can be viewed with a variety of software. We also
include Excel and MATLAB routines that illustrate geometric aspects of the two
problems and provide Monte Carlo simulations for estimating probabilities.
We first state two problems in terms of the
probability of making valid choices for points in a circle of radius 1 centered
at the origin (often called a unit circle). Next we discuss how to use
these problems with groups of students.
Assumption: Let A
and B be two points within the unit circle.
Problem 1. Estimate the probability that the
length of the segment from A to B is less than 1,Problem 2. Estimate the probability that a
circle centered at the midpoint of the segment from
the radius of the unit circle.
A to B with radius 1/2 the length of segment from A
to B lies entirely within the unit
The two problems are independent of one another; that
is, in Problem 2 it is not assumed that the length of the segment from A
to B is less than 1.
With students we can should emphasize the algebraic and
geometric concepts needed by anyone trying to solve either of the
problems. This gives a real situation in which concepts play a roll for
determination of a more complicated mathematical problem. The students can
feel that they are participating in the solution process using mathematics to
contribute to a team effort. Some of the tasks required to develop techniques to
solve these problems are listed next.
(a) How do we check that a pair of points lie in
the unit circle?
(b) How do we compute the length of the segment
between two points?
(c) How do we find the midpoint of a line
(d) How do we determine if a circle with radius
1/2 the length of segment from A to B lies entirely within the
Techniques for answering (a) and (b) are
used in Problem 1, while solutions for (a) - (d) are required for
- For (a): On the board or in a handout draw
several unit circles with different choices for points A and B.
(Specify the coordinates.) Choose some with both points inside the unit
circle and others with 1 inside and the other outside, or both outside. Let
the class experiment. Guide the class to get a well formulated criteria,
then have them check things.
- Tasks (b) and (c) are just are straight
forward use of formulas that should be familiar or easily looked up.
- For (d): Here is where some pictures will
help, like those in Figures 1 and 2. Let students approximate the midpoint
of the segments from A to B and then sketch a circle of radius
1/2 the distance from A to B centered at the midpoint. See
Figures 3 and 4.