Events

You often hear people speak about the probability of this or that event, don't you? However, in order for this phrase to be mathematically meaningful, it is necessary to define both "probability" and "event". In this topic, we will look at the basic concept of probability theory – the concept of an event.

Events

An event is a central concept of the probability theory. In loose terms, it is anything that has a probability of happening. When we talk about the probability of finishing a marathon or the probability of getting a straight flush in poker, "finishing a marathon" and "getting a straight flush" are events. This should be simple enough, and the mathematical definition of an event is equally uncomplicated.

Simple examples are getting heads in a coin toss or a 2 when rolling a dice. Now, to build an understanding about why it makes sense to represent events as sets, let us analyze one more example in detail. Consider an experiment of drawing a random card from a deck of 52 cards. If we represent each card by a pair $(rank, suit),$ we have the following sample space:$\Omega_{random~card} = \{(i, j) : i \in \{2, ..., 10, J, Q, K, A\}, j \in \{\color{red}\diamondsuit \color{black}, \color{red}\heartsuit \color{black}, \spadesuit, \clubsuit \}\}$

$\Omega$ is called the set of elementary outcomes, and its elements are called elementary outcome. In the example above, one of the elementary outcomes is $(2, \color{red}{\heartsuit}\color{black} )$.

Suppose now that we want to describe the event "drawing a black card", which we denote as $E_{black~card}.$ We know what a black card looks like, but what if we wanted to write a computer program to recognize that event? We would need a clear representation of what a black card is, and probably the most intuitive one would be a list of all the corresponding cards in the deck. That may sound too simple, but that is exactly the mathematical way of representing a collection of things that satisfy a certain quality, e.g. being of clubs or spades. In short, we define a set.

$E_{black~card} = \{(2, \spadesuit), ..., (A, \spadesuit), (2, \clubsuit), ..., (A, \clubsuit)\}$The set above is a complete mathematical representation of the event "drawing a black card". Now, for our program to verify whether that event has happened or not, it suffices to check whether the card drawn belongs to the set above or not. To put it slightly more formally, we say that an event $E$ occurs when the outcome of a trial $\omega$ is one of the outcomes in $E$ or, equivalently, when $\omega \in E$.

The truly remarkable insight here is that anything we can say about the result of an experiment can be translated into a set. Think of all the things we could say about that next random card. They all correspond to a set of cards, as in the following examples.

• Drawing a king: $E_{King} = \{(K, \color{red} \diamondsuit \color{black}), (K, \color{red}\heartsuit \color{black}), (K, \spadesuit), (K, \clubsuit)\}$

• Drawing a card of diamonds: $E_{Diamonds} = \{(2, \color{red} \diamondsuit \color{black}), ..., (A, \color{red} \diamondsuit \color{black}) \}$

• Drawing a face card: $E_{face\,card} = \{(i, j) : i \in \{J, Q, K\}, j \in \{\color{red}\diamondsuit \color{black}, \color{red}\heartsuit \color{black}, \spadesuit, \clubsuit \}\}$

• Drawing a queen of spades: $E_{queen~of~spades} = \{(Q, \spadesuit)\}$

• Drawing a card: $E_{a~card} = \Omega_{random~card}$

There are two things worth noticing here.

1. An event is always a subset of a sample space, that is, $E \subseteq \Omega.$ That is not surprising, as every event is described by a set of outcomes, and a sample space contains all possible outcomes. In particular, the event "something happens", or "drawing a card" in our example, coincides with the sample space.

2. An event is always a set. Even when we are concerned with a single outcome, such as $\omega = (Q, \spadesuit)$, the corresponding event is not the outcome itself but a set containing it, that is, $E = \{(Q, \spadesuit)\}.$ In those cases, we speak of a simple or elementary event.

Sure and impossible events

We have just learned that an event is a subset of a sample space. From that definition, we derive two specific events that exist for every experiment and are worth mentioning.

First, remember that a set is always a subset of itself. Therefore, the sample space $\Omega$ is itself an event. It is an event that always occurs because it contains every possible outcome by definition. That is why we call $E = \Omega$ a trivial or sure event.

Similarly, the empty set $\varnothing$ is a subset of $\Omega$. It constitutes an event that can never occur because, being empty, it contains no outcomes by definition. We call $E = \varnothing$ an impossible event.

Combining events

One important aspect of events is that they are not necessarily mutually exclusive. Simply put, events can occur simultaneously. That is probably clear in the previous examples, but let us consider another one.

Consider a game of pool with 15 numbered balls. If we are interested in which of the 15 balls will be the first one to fall, our sample space $\Omega_{\#1}$ would be a set with all the balls in the game:$\Omega_{\#1} = \{1, 2, 3, ..., 15\}.$ We know the outcomes of an experiment must be mutually exclusive, as they are here if we assume only one ball can be the first. However, there is no such constraint on events. For instance, the events "potting the ball #2", "potting an even ball" and "potting a prime ball" do happen at the same time if the outcome is $\omega = 2$.$E_{\#2} = \{2\} \quad\quad\quad E_{even} = \{2,4,6,8,10,12,14\} \quad\quad\quad E_{prime} = \{2,3,5,7,11,13\}$That means it makes sense to talk about a union and an intersection of events. A union of two events $E_1 \cup E_2$ is an event that occurs whenever either of events $E_1$ and $E_2$ occurs. $E_{even} \cup E_{prime} = E_{even~or~prime} = \{2,3,4,5,6,7,8,10,11,12,13,14\}$An intersection of two events $E_1 \cap E_2$ is an event that occurs whenever both events $E_1$ and $E_2$ occur. $E_{even~and~prime} = E_{even} \cap E_{prime} = \{2\}$An intersection of two events might be empty, which means they do not occur simultaneously. For instance, $E_{\#2} \cap E_{\#3} = \varnothing$ because the first ball cannot be both a ball number 2 and a ball number 3, which results in an impossible event. In those cases, we talk of mutually exclusive events.

We can go on and extend this idea to any number of events. If $E_1, E_2, ... , E_n$ are events, their intersection and union are also events. $E_{union}$ is the event that occurs whenever any of the $n$ events occurs, and $E_{intersection}$ is the event that occurs when all $n$ events occur.$E_{union} = \bigcup_{i=1}^{\infty}E_i \quad\quad\quad E_{intersection} = \bigcap_{i=1}^{\infty}E_i$

Conclusion

Now we have learned the concept of an event. Firstly, we introduced the set of elementary outcomes $\Omega = \{ \omega_1, \omega_2 ... \}$, where $\omega_i$ is an elementary outcome, and defined any subset of $\Omega$ as an event. Now, if we know that an event is just a set, we can consider the union and intersection of events simply as corresponding operations on sets. We have also introduced an important term: mutually exclusive events. Events are mutually exclusive if their intersection is empty, which means that they can't happen at the same time.