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Composition allows us to build new ideas using our existing ideas. By learning about probability spaces, ideas in statistics become composable, which means we can build with them.
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We wrapped up last class by clarifying the distinction between a set and a subset. A set is a collection of things - and we really mean things. It could be a collection of words, images, computers, flowers — essentially anything. We denote sets via $\{\}$. A subset is a set that contains some of the elements of the original set. Check out the examples below:
Examples of Sets and a Subset. Note there are multiple possible subsets for each of these sets. I have only listed one subset for each set.
As we mentioned previously, when analyzing something that is uncertain, it’s not enough to simple list out all of the things that can occur. For instance, if we’re talking with our financial advisor, we expect them to provide us with more “information” than just saying that the price of Tesla’s stock tomorrow at 12pm is in $[0, \infty)$. We expect them to provide us with some information about the likelihood of certain values.
To introduce probability into our framework, we’ll make use of a function known as a probability measure. Denoted by $\mathbb{P}$, a probability measure assigns probability to subsets of the sample space. That is, given a subset of the sample space $A \subset \Omega$, the probability measure can tell us how likely that subset is to occur: $\mathbb{P}(A)$.
For example, let’s say that we’re playing darts with our friend and our friend always hits the dart board. The sample space would then be the entire dart board, and a probability measure would tells us the probability that certain subsets of the dartboard occur. A subset is said to occur if the realized outcome lies within the subset.
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Check Your Understanding
Let’s think about to the rolling of a dice example. Why is $\mathbb{P}(3)$ not valid, but $\mathbb{P}(\{3\})$ is valid. Mathematically, what’s the difference between $3$ and $\{3\}$?
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Let’s say that our dart playing friend wins a dollar if the dart is within 4 inches of the center point and otherwise gets nothing. We’ll denote this mathematically with as a random variable $X$. And let’s further assume - for the moment - that the probability measure is known. That is, we know the function that tells us the probability of certain subsets occurring. How might we reason about the probability that our friend wins a dollar?
Let’s be extremely literal for the moment. We know that a probability measure tells us the probability of a subset of the sample space. The set of interest, $\{1\}$ , is a subset so that part checks out. But, it’s not a subset of the sample space so we can’t call $\mathbb{P}(\{1\})$. So what should we do?
Let’s think about under what conditions does the set, $\{1\}$, occurs? We’ll our friend gets a dollar if their throw is within four inches of the center of board. So getting a dollar is essentially equivalent to having the dart land within four inches of the center. So the probability of getting $\{1\}$ must be the same as the probability of the dart landing within four inches of the center. And not only that, but “within four inches of the center” is a subset, so we can assign probability to it!