Teaching Probability

Here is a crude representation of what I use as a visual aid for explaining probability: Unit Square.

Boringly simple, isn’t it?

It has some nice properties. Let’s have an event A that will happen with probability ℙ(A).

Notice that both the length of the shorter edge of the red region and the area of the red region are equal to ℙ(A). While nice and useful, this is simultaneously also horrible as it stretches your brain to think about probability as a length (one-dimensional) and as an area (two-dimensional), which it is neither of, since it is just a ratio (dimensionless). Anyway, if you stomach that, you can use it to draw pictures that can help you reason about probabilities.

Let’s look at a demonstrative example in the following picture.

From that we can easily write down the following two formulas:

$$\mathbb{P}(B|A)=\frac{\mathbb{P}(A\&B)}{\mathbb{P}(A)}\mathbb{P}(A|B)=\frac{\mathbb{P}(A\&B)}{\mathbb{P}(B)}$$

And from there through

$$\mathbb{P}(A\&B)=\mathbb{P}(A|B)\cdot \mathbb{P}(B)$$

we get straight to

$$\mathbb{P}(B|A)=\frac{\mathbb{P}(A|B)\cdot \mathbb{P}(B)}{\mathbb{P}(A)}$$

Boom: Bayes’ theorem. 🖐️🎤💥