Lecture 1: Sample Spaces and Events

What is Probability?

A probability is a number between 0 and 1
It measures how likely an outcome is
P(\Omega) = 1 — something always happens
P(\emptyset) = 0 — the impossible never happens

Welcome to Probability Fundamentals. I want to start with a question that sounds almost too simple: what exactly is a probability?

At its core, a probability is a real number assigned to an outcome or a collection of outcomes. That number always lives between zero and one, inclusive. Zero means the outcome is impossible — it will never happen under any circumstances in our model. One means the outcome is certain — it will always happen, no matter what. Everything in between represents some degree of likelihood. Half means equally likely to occur or not. Point nine means very likely. Point zero one means very unlikely but not impossible.

Now two foundational rules follow immediately from this. First: the probability of the entire sample space — the set of all possible outcomes, denoted capital Omega — is exactly one. This says that something must happen. You cannot run an experiment and have no outcome occur. Second: the probability of the empty set — the set containing no outcomes — is zero. There is nothing in the empty set, so it can never occur.

These two rules seem obvious, but they are the bedrock upon which the entire mathematical theory of probability is built. Everything else — conditional probability, independence, expectation, the Black-Scholes formula — rests on these simple foundations.

One more thing I want you to appreciate from the start: probability is a model. It is a mathematical object we use to represent uncertainty about the real world. Whether that model is a good fit for reality is a separate question — one that statistics and econometrics try to answer. But within the model, the rules are precise, unambiguous, and logically airtight.

Sample Spaces

The sample space \Omega is the set of all possible outcomes
Coin toss: \Omega = \{H, T\}
Die roll: \Omega = \{1, 2, 3, 4, 5, 6\}
Stock price tomorrow: \Omega = (0, \infty)

The sample space is perhaps the most important concept we introduce today, because every probability model begins with it. The sample space — written Omega — is the complete, exhaustive list of every outcome that could possibly occur when we run our experiment or observe our random phenomenon.

Let us look at three examples, ordered from simplest to most complex.

The first is a coin toss. Only two outcomes are possible: Heads or Tails. The sample space has exactly two elements. This is a finite sample space.

The second is a die roll. A standard six-sided die can land on any face from one to six. The sample space has six elements. Again, finite.

Now here is where things get interesting. What is the stock price of a company tomorrow? In principle, the price could be any positive real number — fifty dollars, fifty point seven three, a hundred and twelve. We model this with the sample space being the open interval from zero to infinity, meaning all strictly positive real numbers. This sample space is not just infinite — it is uncountably infinite. You cannot list its elements one by one. There is no enumeration that catches every real number.

This distinction has deep consequences. Finite and countably infinite sample spaces are handled with sums. Uncountably infinite sample spaces require integration. When we define expectation in lecture two, you will see exactly how this plays out.

I also want to flag something important: choosing a sample space is a modeling decision, not a mathematical one. You must think carefully about what outcomes are relevant to your problem. In the Black-Scholes model, stock prices are always strictly positive — zero is excluded by assumption. In other models, zero might be allowed. Different choices lead to different mathematics.

Events and Set Operations

An event is any subset A \subseteq \Omega
Union A \cup B — A or B occurs
Intersection A \cap B — A and B both occur
Complement A^c — A does not occur

Once we have the sample space, we can define events. An event is any subset of the sample space — a collection of outcomes we group together because they share some property we care about.

For example, with a die: the event “the die shows an even number” is the subset containing two, four, and six. The event “the die shows a number greater than four” is the subset containing five and six. Both are perfectly valid events. The full sample space itself is an event — the certain event. The empty set is an event — the impossible event.

Now we combine events using three fundamental set operations. Learn these deeply — they will appear every single day for the rest of the course.

The first is the union, written A union B. An outcome belongs to A union B if it belongs to A, or to B, or to both. In plain language: the union represents the event that at least one of A or B occurs. If A is “it rains tomorrow” and B is “it is windy tomorrow,” then A union B is “it rains or it is windy or both.” The probability of a union is generally not simply P of A plus P of B — we need to subtract the probability of the intersection to avoid double counting. You will see this in the inclusion-exclusion principle.

The second is the intersection, written A intersect B. An outcome is in A intersect B only if it belongs to both A and B simultaneously. This represents A and B occurring together. In our example: it rains AND it is windy. When A and B cannot occur simultaneously — they share no outcomes — we call them mutually exclusive, or disjoint.

The third is the complement, written A-superscript-c. The complement of A contains every outcome in the sample space that is not in A. It represents A not occurring. A critical identity: P of A-complement equals one minus P of A. This is used constantly when the complement is easier to compute than the event itself.

These three operations — union, intersection, complement — are the grammar of probability. Every statement we make about events can be expressed using exactly these three building blocks. And the Kolmogorov axioms — the three mathematical rules that govern how probabilities are assigned — tell us precisely how probabilities interact with these operations. That is the logical foundation of everything we will do in this course.