Definition of Probability

What does it mean when someone makes a claim like "the probability that you find a pearl in an oyster is 1 in 5,000?" or "the probability that it will rain tomorrow is 52%"?

Events and Experiments

When we speak about probabilities, there is always an implied context, which we formally call the "experiment". For example: flipping two coins is something that probability folks would call an experiment. In order to precisely speak about probability, we must first define two sets: the set of all possible outcomes of an experiment, and the subset that we consider to be our event (what is a set?).

[todo] In the world of probability, events are binary: they either happen or they don't.

Definition of Probability

It wasn't until the 20th century that humans figured out a way to precisely define what the word probability means:

$$ \p(\text{Event}) = \lim_{n \rightarrow \infty} \frac {\text{count}(\text{Event})} {n} $$

In English this reads: let's say you perform $n$ trials of an "experiment" which could result in a particular "Event" occurring. The probability of the event occurring, $\p(\text{Event})$, is the ratio of trials that result in the event, written as $\text{count}(\text{Event})$, to the number of trials performed, $n$. In the limit, as your number of trials approaches infinity, the ratio will converge to the true probability. People also apply other semantics to the concept of a probability. One common meaning ascribed is that $\p(E)$ is a measure of the chance of event E occurring.

Measure of uncertainty: It is tempting to think of probability as representing some natural randomness in the world. That might be the case. But perhaps the world isn't random. I propose a deeper way of thinking about probability. There is so much that we as humans don't know, and probability is our robust language for expressing our belief that an event will happen given our limited knowledge. This interpretation acknowledges your own uncertainty of an event. Perhaps if you knew the position of every water molecule, you could perfectly predict tomorrow's weather. But we don't have such knowledge and as such we use probability to talk about the chance of rain tomorrow given the information that we have access to.

Origins of probabilities: The different interpretations of probability are reflected in the many origins of probabilities that you will encounter in the wild (and not so wild) world. Some probabilities are calculated analytically using mathematical proofs. Some probabilities are calculated from data, experiments or simulations. Some probabilities are just made up to represent a belief. Most probabilities are generated from a combination of the above. For example, someone will make up a prior belief, that belief will be mathematically updated using data and evidence. Here is an example of calculating a probability from data:

Probabilities and simulations: Another way to compute probabilities is via simulation. For some complex problems where the probabilities are too hard to compute analytically you can run simulations using your computer. If your simulations generate believable trials from the sample space, then the probability of an event E is approximately equal to the fraction of simulations that produced an outcome from E. Again, by the definition of probability, as your number of simulations approaches infinity, the estimate becomes more accurate.

Probabilities and percentages: You might hear people refer to a probability as a percent. That the probability of rain tomorrow is 32%. The proper way to state this would be to say that 0.32 is the probability of rain. Percentages are simply probabilities multiplied by 100. "Percent" is Latin for "out of one hundred".

Axioms of Probability

Here are some basic truths about probabilities that we accept as axioms:

These three axioms are formally called the Kolmogorov axioms and they are considered to be the foundation of probability theory. They are also useful identities!

You can convince yourself of the first axiom by thinking about the math definition of probability. As you perform trials of an experiment it is not possible to get more events than trials (thus probabilities are less than 1) and it's not possible to get less than 0 occurrences of the event (thus probabilities are greater than 0). The second axiom makes sense too. If your event is the sample space, then each trial must produce the event. This is sort of like saying; the probability of you eating cake (event) if you eat cake (sample space that is the same as the event) is 1. The third axiom is more complex and in this textbook we dedicate an entire chapter to understanding it: Probability of or. It applies to events that have a special property called "mutual exclusion": the events do not share any outcomes.

These axioms have great historical significance. In the early 1900s it was not clear if probability was somehow different than other fields of math -- perhaps the set of techniques and systems of proofs from other fields of mathematics couldn't apply. Kolmogorov's great success was to show to the world that the tools of mathematics did in fact apply to probability. From the foundation provided by this set of axioms mathematicians built the edifice of probability theory.

Provable Identities

We often refer to these as corollaries that are directly provable from the three axioms given above.

This first identity is especially useful. For any event, you can calculate the probability of the event not occurring which we write in probability notation as $E\c$, if you know the probability of it occurring -- and vice versa. We can also use this identity to show you what it looks like to prove a theorem in probability.