**Objective**:
To
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.
**Level: **
Algebra, precalculus, discrete math, courses involving
elementary probability, or introductory programming courses.

**Prerequisites:
**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.

**Platform: **
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.

**Instructor's
Notes:**

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,

the radius of the unit circle.

**Problem 2.** Estimate the probability that a
circle centered at the midpoint of the segment from

**A** to **B** with radius 1/2 the length of segment from **A**
to **B** lies entirely within the unit

circle.

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
segment?

**(d)** How do we determine if a circle with radius
1/2 the length of segment from **A** to **B** lies entirely within the

unit circle?

Techniques for answering **(a)** and **(b)** are
used in Problem 1, while solutions for **(a)** - **(d)** are required for
Problem 2.

__Suggestions__

- 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.