This page first describes and discusses a problem
and then allows the user to do a simulation of a model of the problem to
arrive at an answer.
Problem Statement
Given that a glass rod is dropped, and that it breaks in exactly
two places (yielding three pieces of rod), what is the probability
that those pieces can be arranged, end to end, to form
a triangle?
Discussion
First, is it possible to form a triangle? Yes, certainly if the rod breaks
into approximately equal pieces. Second, do the pieces have to form a triangle?
No, because we could get two extremely short pieces and one long one. Therefore, either
outcome can happen.
Is there a way to tell, given the lengths of the pieces, whether or not they form a triangle?
What do we know about a triangle? We know that the length of any one side must be
less than the sum of the lengths of the other two sides. (In running the simulation given below,
see if you can identify an easier method for determining if the pieces form a triangle.)
Solution
The mathematics needed to solve this is not generally familiar. How about
trying another approach. How about dropping a large number of glass rods. Then we
will examine only those cases where the rod breaks in two places. For those cases, we
will keep track of the number of cases and the number of times that the pieces can form
a triangle. Then, an approximation to the probability will be the proportion of
successes out of the number of cases. But this will be a messy task. How about letting
the computer simulate this.
Simulation
The computer has a random number function. We can use it to
determine just where the two breaks will happen. First, we set the length of the rod.
We will use 10 centimeters. Then, we will select two random breaks, i.e., numbers between 0 and 10.0 centimeters.
This gives us the three pieces. Then we determine if the pieces can form a triangle.
And, we keep track of the number of rods dropped, the number of times that the pieces
do form a triangle, and the proportion of success.
Shortcomings
The proportion is not a mathematical solution. It is at best an approximation.
Each time that we do the simulation, in fact, each time that we drop another rod, the
proportion changes. The true answer is a constant. There is a probability. This method
will produce an approximation, but it will not give a complete solution.