The triangle inequality and geometric probability

I finished up a section on the triangle inequality today with my older son and wanted to show him a surprising application. This example – taken from geometric probability – is a little over his head, but was still fun to talk through.

The problem is this:

You have a stick with length 1. If you chop the stick into 3 pieces, what is the probability that those three pieces can form a triangle?

Fun question – we started out with him giving his thoughts on the problem, and the problem of thinking through probability when you have an infinite number of choices:

With the initial thoughts out of the way, we started thinking through a few specific ways to cut the stick into pieces. The important observation from this part of the talk is that if any piece has length greater than 1/2, we won’t be able to form a triangle:

Finally we get to the geometric probability. We have to start with a special property of an equilateral triangle. We’d talked about this property of an equilateral triangle yesterday. My son remembered the property, but not the proof. We spent a minute or two reviewing the proof, but the proof wasn’t the main point.

We can use an equilateral triangle with height of 1 to model our probability problem in a surprising way. With our new model, we see that the amazing connection to geometry tells us that the probability of our broken up stick forming a triangle is 1/4.

So, a fun little project that is a little more advanced than what we’ve covered so far. It touches on geometric probability which is an incredible subject on its own, it touches a little on properties of infinity, and finally the relationship between algebra and geometry.

All from the simple ideas behind the triangle inequality. Amazing!