Models

Models

We will learn how to create simple models that enable us to gain insight into the problem we are investigating. You also usually can not estimate your quantity directly, but must instead break down your estimate into other smaller estimation problems that can be tackled individually. We will start with simple estimations and then move on to estimations that require more planning.

Here are a few ideas about models:

• A model is a simplified representation of a problem we'd like to solve

• A model is an approximation that allows us to think about the world

• Models:

  • rectangle

  • linear increase or decrease

  • exponential increase or decrease

  • a roll of a die or several dice

  • a bathtub with a faucet and a drain

• The situation you are modeling won't always look exactly like the model

• All models are wrong, some are useful

• Models often contain assumptions about the problem that others may not share

Estimations and models

Here are some steps that can be useful as you develop models:

• Understand the problem

• Take a guess

• Identify the most important quantities

• Detail the assumptions you are making

• Estimate the value of these quantities

• Set up the computation as clearly as possible

• Execute the computation

• Decide if the result is plausible

• See if it matches your original guess

More complex models

In the Art of Insight, Sanjoy Mahajan presents ways of determining what estimations you need to make in your model. A useful tool is a tree model.

This diagram is a tree model to help us determine the volume of a dollar bill.

Tree Model

Note how we break up our goal estimation into smaller problems and then work to determine if the smaller problems require further decomposition.