Geometry Portal

Objective 1c: Monte Carlo Method

You will be able to use lengths, areas, and volumes to find geometric probabilities

You’re probably familiar with π as something you have to multiply by to get the area or circumference of a circle. The more sophisticated student will tell you that π is the ratio of any circle’s circumference to its diameter. Of course, such a ratio is not easy to measure accurately with a tape measure or a ruler, so mathematicians have developed alternative methods for calculating digits of π. One such technique is called the Monte Carlo Method, named after a city renown for its gambling casinos, which is essentially based on geometric probability.

Monte Carlo Method

The first approximation of pi as 3.14 was calculated a couple thousand years ago by Archimedes, the same guy who famously (and nakedly) shouted "Eureka" after discovering a method to measure the density of an object while bathing. While he used a method called exhaustion for his approximation, we will use a Monte Carlo Method based on probability. Duration: 6:28

Experimental vs. Theoretical Probability

Still confused as to how we will use the previous example can be used to approximate the value of π? Well, the answer has to do with the relationship between the theoretical and the experimental probability of an event. That, and multiplying by 4. Duration: 5:29

Approximating Pi Box

This activity requires a special set up that can be duplicated with the print out below. If you have the resources, click or tap the image below to download a copy of the Approximating Pi box. I would suggest printing the box template out on two separate pieces of cardstock. Once assembled, you will also need a pile of approximately 30 small beans. Lentils will work well. Remember that you are conducting 5 trials and recording your results.

Approximating_Pi_Box

Activity Time-Lapse

If you need some guidance creating your box, watch this brief time-lapsed video. Try not to blink. Duration: 0:24

Monte Carlo Applet

https://youtu.be/XrrrFk46rlgTo conduct this experiment virtually, use the embedded applet below. First, input a number in the box for N. This basically represents the number of beans in each trial up to a maximum of 10,000. As soon as you hit enter, your chosen number of "beans" will be thrown and an approximation of pi calculated. Click the right arrow with an "x" on it to re-throw your "beans". Hitting the arrow bent into a partial circle resets the simulation.

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