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 $\mathbb{P}(A)$.

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

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

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

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

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. 🖐️🎤💥