In This Set of Notes We’re Going to
In the text below, you will encounter a number of new concepts that will make you think that you don’t know that much about statistics and probability.
I’d like to argue, at the outset though, that you likely know quite a bit! You know that the probability that a dice lands on a 2 is the same as the probability that it lands on a 3 which is $1/6$. You know that the probability that a coin comes up heads is $1/2$. You know that no one knows the probability that Tesla’s stock is above $300 tomorrow. And you know that the probability that someone gets either an A or a B in a class is higher than the probability that the person receives an A. You know a bunch of things about probability.
The focus of this note is to introduce a framework for reasoning about uncertainty so that you can understand the relationship between these things that you already know. Having a framework will allow you to better process new information and tackle new problems.
The framework is called a probability space. It consists of three components.
A framework for thinking about uncertainty has to represent two pieces of information. (1) What are all the things that could happen. We’re uncertain about something. What are all the things the could happen. And (2), how likely are certain things to happen. Maybe everything is equally likely. Maybe certain things are more likely than others. A framework that allows one to think about uncertainty needs to be able to represent these two pieces of information.
To make things clear, let’s use the game of darts as a working example for the rest of this note. Throwing a dart is an uncertain process. You throw the dart and it lands somewhere on the board (we’ll assume in this discussion that you always hit the board). You throw the dart again and it lands likely somewhere else on the board. Each time maybe your arm angle is a little bit different. The weight on your feet is sometimes more forward. Your knees bend more when the background music is really bumping — https://www.youtube.com/watch?v=t7bQwwqW-Hc
We can introduce a framework for reasoning about this uncertain process of throwing darts by (1) describing the set of all possible outcomes — the dartboard in this example. And (2) the intuitive idea of relatively likelihood can be represented by assigning a value between [0, 1] to each region of the dartboard to indicate the probability that the dart lands in that region.
We refer to the set of all possible outcomes, as the Sample Space, and denote it by $\Omega$. It’s the dartboard in our example. When you through the dart it lands on some point in the dartboard. We use $\omega$ to denote an element in the sample space.