The language used to develop probability distributions.
The discussion starts with the idea that we have a space which holds some number of potential outcomes.
The space is usually defined as Omega or $\Omega$. A simple example of this concept is the possible outcomes when rolling a six-sided die.
$$ \Omega = \lbrace 1,2,3,4,5,6 \rbrace$$
There can be any number of sets and it can get rather large quickly. For instance if we were watching a horse race where 10 horses raced. The set of possible outcomes is the number of different orders that the horses could have finished in is $10!$ or 3,628,800 different combinations.
There are also sets that are called countably infinite so that $\Omega = \lbrace x_1, x_2, x_3, … \rbrace$.
And then there is the simplest set known as the empty set or $\theta$.
Now inside this set $\Omega$ we are going to “assume” there is a subset of measurable events $\mathcal{S}$ inside $\Omega$ that can have probabilities assigned so that each event $\alpha \in \mathcal{S}$ is a subset of $\Omega$.
So in the case of a six-sided die, if we rolled a 6 then that’s an event $\lbrace 6 \rbrace$. All the chances for rolling an odd number would be $\lbrace 1,3,5 \rbrace$.
Now if we had 10 horses racing, what would be the odds that Tacitus would win the Breeders’ Cup Longines Classic?
To generalize the idea of an event space it needs to satisfy 3 things:
First the event space must include the empty set $\theta$.
Second it must
- It is closed under union.
- It is closed under complementation.
The best way to show the latter two are with Venn diagrams. Stop here on that.
Interpretation of probability
I think this is the fascinating part that I wasn’t aware of and starts to get at what intuitions are we talking about.
Consider $\mathcal{P} ( \alpha )$ of an event $\alpha$ puts a degree of confidence on whether $\alpha$ will occur.
But what does the actual number represent? The most common interpretation comes from the frequentist framing in terms of the frequency of events. In other words it’s the fraction of times it occurs relative to the number of total outcomes if done indefinitely.
Take the die rolling example of rolling an odd $\mathcal{P} (\alpha) = 0.3$ for $\alpha = \lbrace 1,3,5\rbrace$. So if we roll the die repeadetly we would expect to get a 0.3.
The frequentist view doesn’t work however when we discuss matters such as whether it will be sunny tomorrow. The frequentist view doesn’t explain how we got to a 30% chance of Sun tomorrow. Rather is a subjective degree of belief. It’s