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Bayes' Theorem

Chapter 1: Bayes’ Theorem for Dummies

If you are confused with the concept of Bayes’ Theorem, this is a fantastic place to start. Before we dive in too far, let’s take a look at Bayes’ Theorem without using the formula.

To begin, let’s draw a rectangle. Don’t get hung up on the shape – it could be any shape, but a rectangle is easy to work with. The area inside the rectangle represents all possible outcomes for our experiment. For example, if there are two possible outcomes that are equally probable, we would divide the rectangle into two equal halves. This means that each outcome has a 50% likelihood of occurring. Or, if there are three possible outcomes that are all equally probable, we would divide the rectangle into equal thirds. This would mean that each outcome has a 33.33% chance of occurring.

For this example, we are going to stick with two equal outcomes and title each outcome A and B, respectively.

Now, imagine that each probability represents a small cardboard box. Box A is filled with 10 chocolate chip cookies. There is nothing else in Box A except home baked, warm, mouth-watering chocolate chip cookies. To demonstrate this, we will shade in Box A. In Box B there are also cookies, but there are two different types. There is an even mix of 5 peanut butter cookies and 5 chocolate chip cookies. To demonstrate this, we will draw a line and cut the box in half, and then shade in the chocolate chip cookies in the bottom half of the rectangle. We will leave the top half blank.

Let’s step back and look at the rectangle. Can you see the chocolate chip cookies in the shape of an L? Those areas represent all of our chocolate chip cookies in both boxes, while the white area represents the peanut butter cookies.

Now, what if you were to close your eyes and have both boxes placed in front of you and shuffled, and then reach out your hand and select a cookie? Let’s say you did this and when you opened your eyes, you saw that you had selected a chocolate chip cookie.

If you had to guess what box the cookie came from, what box would you select? Many people would select Box A, and we’ll take a hunch that you are one of them. But let’s take a closer look at why this is. Both Box A and B have chocolate chip cookies, but Box A has exactly double the number of chocolate chip cookies than Box B. Within a split second your brain assessed this and came away with the conclusion that Box A has a greater probability of being selected than Box B.

Within a split second, you quickly became more confident in one probability versus another. One box versus another. And then, you made a decision.

Here’s the magic! This calculation is a very basic, natural use of Bayes’ Theorem. Given evidence (the amount and type of cookies in each box), you were able to quickly come to the conclusion that Box A has a greater probability of being selected than Box B.

Now, let’s step back once more. When your hand-selected a chocolate chip cookie, something disappeared: the probability of selecting a peanut butter cookie is now gone. So, to visualize this, let’s wipe away the portion of Box B that represents the peanut butter cookies.

Our boxes are now in the shape of an L, and we can also see that there are double the number of cookies in Box A than Box B. In fact, if we were to break the Boxes apart into equal sections, we would have 3 areas: 2 sections in Box A, and 1 section in Box B.

By looking at this, we can see that Box B has a probability of ⅓, or ~33% of being selected. Box A has a probability of ⅔, ~66% of being selected. This difference in probability is what your brain roughly calculated before and the whole reason why you selected Box A. Your Brain looked at ~33% vs ~66% and selected the highest percentage, which comes from Box A. * The ~ symbol means approximately.

What we have just done is demonstrate the concept of Bayes’ Theorem and solve a problem all without using the formula. Now, before we solve this same problem with the formula, it might be helpful to define the formula and its components, or as we call them ingredients. Go to Chapter 2: Bayes’ Theorem Formula: A Simple Overview

Continue on to Chapter 2: Bayes’ Theorem Formula: A Simple Overview.