# Lesson-planning: Teaching probability concepts via geometry

I am intending to teach a lesson covering some topic related to "Probability via Geometry" and, if possible, I would appreciate references or materials (or good ideas) that can help me. The target students for this lesson are between 14 and 17 years old, so I would like to talk about basic results, presented as intuitively as possible, and for which any proofs can be understood by the audience.

The subject of this class is described as: "strategies to teach probability concepts via geometry."

Question: Does anyone have references, materials, or other ideas about teaching probability via geometry for students age 14-17?

• Young guys with ages between 14 and 17 years old without special background in formal disciplines? – Incnis Mrsi Aug 19 '15 at 18:53
• Any kind of geometry in particular? – DavidButlerUofA Aug 19 '15 at 19:39
• Is this about a single lesson? A unit/module? An entire course? Are you looking for an individual task that incorporates ideas from geometry and probability? E.g., computing the probability that at a random time in the day the smaller angle (measured in radians) between the hour and minute hand is $\leq \pi/2$? – Benjamin Dickman Aug 20 '15 at 0:51
• How about the classic question: Remove two points in $[0,1]$ at random. What is the probability that the resulting three line segments can be sides of a triangle? This is clearly a probability question, and solving it (at least in the classic way) involves a bit of geometry: (1) talking about the importance of the triangle inequality - sufficient and necessary for a triangle to be formed; (2) graphing and shading to solve the problem. You could also pose the follow-up about five points forming a tetrahedron; that problem turns out to be tough! – Benjamin Dickman Aug 20 '15 at 15:51
• One concept is that probability problems can be represented geometrically: As mentioned before, the graphing and shading technique can be used here. If they are 14-17, then maybe it makes sense for you to pick the first point; say, at $1/3$. What is the probability that the second point, picked at random, admits a triangle? Have them pick a different first point, and solve again. Then ask the original question (both points picked at random). This could scaffold nicely towards the full problem, and indicate how probabilities can change depending on the givens (I'm wondering about Bayes...). – Benjamin Dickman Aug 20 '15 at 16:06

I see at least one good topic for this: Monte-Carlo algorithm to compute area or volume. It has a strong geometric side, is definitely probabilistic, only involves simple concepts, and is even relevant to real-life (of mathematicians).

Imagine we want to compute the area of a region in the plane, say a oddly-shaped pool in a fancy California hotel. Problem is, we are lazy and currently staying at the 20th floor's balcony of our room.. Good news is, we have a very large number of beers available in the minibar, we know the area of the rectangular garden where the pools lies and the pool is currently closed so we do not risk harming people in what follows.

Let simply drop randomly beer cans in the garden and count the amount of them that fall in the pool. If we divide this by the number of cans which did not fell out of the garden altogether, assuming we dropped them somewhat uniformly, we should get a good estimate of the ratio between the areas of the pool and the garden. Then we can deduce the former by multiplying the later by the above ratio.

I phrased it in a quite unscientific context, but the point is that this kind of method turns out to be indeed useful in applied maths to compute areas, or more generally higher-dimensional volumes.

Note that this method becomes unusable in high dimensions because irregular shape tend to occupy an exponentially small fraction of the least cube they are contained in. There are other methods available then, also using probabilities (randomly bouncing billiards, hit and run) which are implemented in computer to obtain volume estimates. In general, it is a hard problem to estimate the volume of a convex set in $\mathbb{R}^n$ given by linear constraints or as convex hulls of a given set of points, but I guess we are getting aside your current issue here.

Here are some examples, many of them gleaned from this link. Note that this link has slides of an entire unit and would probably be very useful.

1. Find the probability of landing on different colors in a spinner. To make it interesting, use different sizes for the different colors. Probability is based on central angle size. Students can also design spinners given different probabilities.
2. Given a target and an arrow that landed on a target, find the probability that the arrow landed in a certain section. This probability would be based on area of concentric circles. This example is a part of a youtube video.
3. Find the probability that three points will be collinear given a picture with points: some on the same line, some outside the line.
4. Given a rectangle with different shapes inside, find the probability that a point chosen will be inside a given shape. Probability is based on angle size.
5. Represent a non-geometric probability problem geometrically. For example, find the probability that a traffic light will be green when you get to the corner. Represent the times as a line segment divided into smaller parts. Each part has a length corresponding to the number of seconds the light will be that particular color.

The classic: what is the probability that two points chosen at random from the interval $[0,1]$ will sum to at least $1$?

You might look at this "module" published by COMAP, The Consortium for Mathematics and Its Applications:

http://www.comap.com/product/samples/6659.pdf

It can be powerfully impressive to approximate $\pi$ by dropping matchsticks on a ruled surface:

Wikipedia image by McZusatz.

Here, $17$ sticks were tossed, the separation between the lines is the stick length, and $\pi \approx (2 \cdot 17) / 11 = 34/11 \approx 3.1$. See, e.g., this description.