Events

We noted previously that the point of defining experiments and sample spaces is to be able to attach probabilities to the different possible events that can occur. Now it's time to say what we mean by "events". In this section we'll discuss events, how to represent them graphically with Venn diagrams, and how to combine them to create new events.

What is an event?

At a first approximation, we can define an event as follows:

Rough definition. Given a sample space \(\Omega\), an event is any subset of \(\Omega\).

But note...

This definition isn't quite correct. To be exactly correct we'd need to invoke the notion of a \(\sigma\)-algebra ("sigma algebra"), but a full discussion is out of scope for an elementary course on probability theory.

For the curious, the idea behind using a \(\sigma\)-algebra here is to guarantee that every event is "measurable", which in the context of probability theory means that we can assign a probability to any event.

In practice, every set we ever encounter is going to be measurable (non-measurable sets are exotic), so we can get away with thinking of events as arbitrary subsets of \(\Omega\). Just know that this is a slight simplification of the truth.

Let's consider some examples to make things clearer.

Example: Events for rolling two dice

Here are some possible events when rolling two dice. Note that we're assuming that \(d_1\) and \(d_2\) are integers in the range 1-6.

Description Set notation
Rolling "snake eyes" (two ones) \(\{ (1, 1) \} \)
Rolling "boxcars" (two sixes) \(\{ (6, 6) \} \)
Rolling an even sum \(\{ (d_1, d_2) \mid d_1 + d_2 \bmod 2 = 0 \}\)
Rolling a sum of 7 \(\{ (d_1, d_2) \mid d_1 + d_2 = 7 \}\)
Rolling the same number on both dice \(\{ (d_1, d_2) \mid d_1 = d_2 \}\)
Rolling a sum of 13 \(\{ (d_1, d_2) \mid d_1 + d_2 = 13 \}\), i.e., \(\varnothing\)

That last one—rolling a sum of 13—may be surprising. There's no way to do that (i.e., no outcome yields that event), and so the event is the null set, \(\varnothing\). But the null set is an event, albeit an impossible event. That is, no matter which outcome occurs, the impossible event never occurs.

Example: Events for spinning a roulette wheel

Here are some possible events when spinning the roulette wheel:

Description Set notation
Landing on red \(\{ 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 \}\)
Landing on black \(\{ 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 \}\)
Landing on 0 or 00 \(\{ 0, 00 \}\)
Landing on a number 1-18 \(\{ p \mid 1 \leq p \leq 18 \}\)

Example: Events for drive time to get to the office

For drive times, here are some possible events:

Description Set notation
Drive time is at least 15 minutes \(\{ m \in \mathbb{R} \mid m \geq 15 \}\)
Drive time is at least 15 minutes but less than 30 minutes \(\{ m \in \mathbb{R} \mid 15 \leq m < 30 \}\)
Drive time is precisely 4.81727331982 minutes \(\{ m \in \mathbb{R} \mid m = 4.81727331982 \}\)
Drive time is at least 0 minutes \(\{ m \in \mathbb{R} \mid m \geq 0 \}\); i.e., \(\Omega\)

Here, once again you may find the last event surprising, since every outcome yields it. That's OK. Ω itself is one of the subsets of Ω and it's called the universal event. That is, no matter the outcome of the experiment, the universal event Ω always occurs. Mathematicians are funny like that.

Now let's look at how we can use Venn diagrams to represent events in a graphical way.

Representing events visually with a Venn diagram

Venn diagrams come from the world of set theory. But since our approach to modeling events is basically to identify them with sets, we can use Venn diagrams to visualize events too.

Let's start with a simple example: the experiment of rolling a single standard die. There are six outcomes in the sample space. We can represent the sample space and its outcomes as follows:

A Venn diagram with no events
A Venn diagram with no events

Here, the entire field is the sample space, and the individual dots are of course the outcomes.

We aren't showing any events yet, but we can fix that. Let's consider the event of rolling a number in the range 1-3. Here's how that looks:

A Venn diagram with a single event
A Venn diagram with a single event

So far so good. What happens if we want to add another event—say, rolling either a 2 or a 5? That's easy to add too:

A Venn diagram with two events
A Venn diagram with two events

That's really all there is to it. Venn diagrams offer a nice way to help see events (or more generally, sets) and the relationships between them. Here, we can see that when the outcome 2 occurs, it's part of both events. The outcomes 4 and 6 aren't part of either event.

Combining events to create new events

In the example above, we described two distinct events that happened to share the outcome 2. We can create new compound events by combining these two events using standard set-theoretic operations.

To simplify the discussion, let's use the following event names:

\[ A = \{ 1, 2, 3 \}, B = \{ 2, 5 \} \]

Then one compound event we can create is the union \(A \cup B\), which is defined as the set \(\{ x \mid x \in A \lor x \in B \}\). Visually, it looks like this:

A Venn diagram with a union event
A Venn diagram with a union event

Another compound event we can create is the intersection \(A \cap B\), which is defined as the set \(\{ x \mid x \in A \land x \in B \}\). Visually:

A Venn diagram with an intersection event
A Venn diagram with an intersection event

A third compound event we can create is the difference \(A \setminus B\), defined as the set \(\{ x \mid x \in A \land x \not\in B \}\). Visually:

A Venn diagram with a difference event
A Venn diagram with a difference event

Obviously we could carry on like this indefinitely, as we can combine existing events (whether simple or compound) into new events using arbitrary set operators. This is important as we will need to have rules for calculating the probability of a compound event from the probabilities of its constituent events.

We'll get to that momentarily. But first let's finish off our discussion of probability spaces by considering the third component,probability measures.

Exercises

Exercise 1. Consider an experiment where you flip three coins. What is the sample space? What are some events you might define for this experiment?

Exercise 2. Define some compound events for the events you identified in exercise 1.