Modeling Experiments

In this section we'll learn about probabilistic experiments and how to model them mathematically.

What's a probabilistic experiment?

Probability gives us a way of modeling probabilistic experiments, and determining how likely various possible results are. But we need to understand both what we mean by a probabilistic experiment and how the model works. Let's start with what a probabilistic experiment is.

Definition. A probabilistic experiment is a procedure that, when executed, results in exactly one outcome. If there is only one possible outcome, the experiment is deterministic. If there are multiple possible outcomes, then the experiment is random.

This probabilistic kind of experiment is different than what you may have studied in science class, and that's why I don't have a photo of beakers or people in lab coats at the top of this page. In the science class type of experiment, known as a controlled experiment, we generally look at the effect of manipulating a single a variable (the experimental variable) while holding all other variables control variables) constant. That way we can evaluate our hypotheses about how the world works.

That's not at all what we're talking about here. Our sense of experiment is much broader: some process produces outcomes, usually random. That's it. It's clearly a very broad notion. Here's a list of examples:

  • flipping a fair coin
  • rolling two dice
  • drawing a card from a standard deck
  • asking 1,000 people which political candidate they prefer in an election
  • counting the number of Xbox sales on Christmas day
  • driving to work and recording the commute time
  • giving 40 adults a vaccine and counting how many show evidence of an immune response

A mathematical model for probabilistic experiments

Mathematically, we use a structure called a probability space to model an experiment. That sounds pretty fancy, and it is in fact very fancy. But hang with me here. I'll explain all of these ideas in gory detail in the following sections.

Definition. A probability space is a triple \((\Omega, \mathscr{F}, P)\) where

  • the sample space \(\Omega\) is the set of all possible experimental outcomes,
  • the event space \(\mathscr{F}\) is a \(\sigma\)-algebra whose elements are subsets of \(\Omega\), and
  • the probability measure \(P\) assigns to each event a probability in the range [0, 1].

Yikes, yikes. But at a high-level, the idea is that to model an experiment, we need to know:

  • what the possible experimental outcomes are (the sample space \(\Omega\) tells us this),
  • what the possible events are (the event space \(\mathscr{F}\) tells us this), and
  • what the probability of each event is (the probability measure \(P\) tells us this).

We'll start unpacking this by investigating the notion of a sample space.