In This Note We’re Going to
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In all probability, Alexander Hamilton is the foremost political figure in American history who never attained the presidency, yet he probably had a much deeper and more lasting impact than many who did. - Source
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In the text below, you will likely 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 the things that you already know. Having a framework will allow you to better process new information and more easily tackle new problems.
The framework, which is known as a probability space, consists of three parts.
Before we explain what each part is (or does), let’s take a step back and think about what information must be included in a framework so that we can reason about uncertainty.
If we think about it for a moment, the framework has to contain two pieces of information. First, it has to tell us what are all the things that could happen. We’re reasoning about uncertainty. We’re uncertain about something. We should start by describing in someway all the things that could happen. For example, if were about to roll a die, we could start by listing out the the set of possible outcomes.
And this set of possible outcomes, what we call the Sample Space and denote using the symbol $\Omega$, is the first part of a Probability Space.