Sample space

Let's say you want to bet on a football (soccer) match. What's the first thing you need to clarify before placing your bet? Of course, you need to agree on what possible outcomes of the match could be. If you're betting on who will win, you'll likely consider three potential outcomes: "team A wins", "team B wins", or "it's a draw". It's the same when you randomly select a contest winner from participants on a specialized website.

Of course, the outcomes you're focused on will depend on the type of bet you wish to place. For example, you could also bet on the number of goals, fouls, or even red cards. In those cases, you'll be focused on completely different aspects of the match, and you'll expect the result to be a whole number.

The first step of any experiment, then, is to give a complete description of all the potential outcomes. In probability theory, this set of all possible outcomes is called a sample space or an outcome space.

Elements of the sample space

As we've just explored, the sample space, denoted as $\Omega$ (Omega), is a set of all possible outcomes of an experiment. All individual outcomes - $\omega$ - are elements of the sample space, $\omega \in \Omega$. We refer to these as sample points. Going back to the football match example, our sample space could be written as a set with the sample points "team A wins" ($A$), "team B wins" ($B$), or "it's a draw" ($D$).

For the contest winners, the sample space will look very similar. For example, if you have $5$ participants: John, Maxwell, Lucy, Emma, and Jane, the sample points can be written as the names of the winners:

Next, let's look at a more complicated example. If you're picking a card from a deck, the sample space has the cardinality of the number of cards. But if you're playing a game where you pick a card from an opponent's hand, then the cardinality of the sample space is the number of cards in that person's hand.

If we go back to football and you want to bet on the number of red cards, our sample space would be quite different. Say the game ends if one team gets 5 red cards; we can have a maximum of $9$ red cards in a match ($4$ for one team, $5$ for the other):

For another example, consider the simplest experiment: flipping a coin. In this case, the sample space consists of two elements, "heads" ($H$) and "tails" ($T$):

If you expand this example to include two identical coins, the sample space would then consist of all combinations of heads and tails: "heads" and "heads" ($\{H\}$), "heads" and "tails" ($\{H,T\}$), and "tails" and "tails" ($\{T\}$). Remember that two sets containing the same elements are regarded as equal. Therefore, we do not differentiate between "heads" and "tails" ($\{H, T\}$) and "tails" and "heads" ($\{T, H\}$). This alignment fits our intuition, as the two coins are indistinguishable, and we couldn't differentiate between these two outcomes.

If you flip the coins one by one, then the order in which you observe the results of the coin flips does matter, and we need to reflect that in our sample space. For this purpose, we can represent the possible outcomes using ordered pairs $(flip_{~1}, flip_{~2})$.

Properties of sample space

As we've seen from earlier examples, we can construct the sample space to match the complexity of the experiment at hand. However, there are two properties that a sample space must adhere to for a consistent probabilistic model.

• Mutually exclusive outcomes: The elements of a sample space must be distinct and mutually exclusive. This means that the outcome of an experiment always has to be unique.

• Collectively exhaustive outcomes: The result of an experiment should always be one of the sample space's possible outcomes. In other words, a sample point $ω$ should always be observed.

With these two properties, we can ensure that each trial of an experiment will result in one-and-only-one outcome.

For example, if you want to extend our football match example to include not only the winner but also the goal difference, we propose the following sample space. Here, $X_{>3}$ means that the team $X$ wins by more than three goals.

Can you see the problem with this sample space? Think about what happens when team A wins $4–⁠0$. In that case, both $A$ and $A_{>3}$ occur, meaning that the outcomes in $\Omega'_{winner}$ aren't mutually exclusive. Now, suppose we try to fix this by suggesting the following sample space.

It's easy to see why $\Omega''_{winner}$ also poses problems. If team A now wins by $1–⁠0$, that wouldn't correspond to any of the outcomes in $\Omega''_{winner}$, meaning that we don't have collectively exhaustive outcomes. The correct solution if you wish to extend our sample space in this way would be to add the complements of $A_{>3}$ and $B_{>3}$, namely A wins by $3$ goals or less $(A_{\leq3})$ and, analogously, B wins by $3$ goals or less, denoted $B_{\leq3}$.

Check if the outcomes in $\Omega'''_{winner}$ are both mutually exclusive and exhaustive. This means that no matter the result of the match, exactly one outcome in $\Omega'''_{winner}$ will be observed.

Can we always find all possible combinations for the sample space? Let's find out together in the next section!

Visualizing sample spaces

Let's say there's a soccer World Cup. As you can imagine, it would be practically impossible to write down all potential results of all games from the group stage to the final. Moreover, in the knockout stage, the outcome of each match influences the outcomes of matches in the next stages.

Here's a picture presenting the round of $16$ of World Cup $2022$:

For example, if Argentina wins against Australia, they'll have to play the winner of the US versus the Netherlands match and so on. So, how many other options are there for that match? Only four: Netherlands vs Argentina, US vs Argentina, Netherlands vs Australia, US vs Australia.

If you're mostly interested in the results of the semifinals, you'll have even more options and a larger sample space. The same goes for the final, as every team has a chance of reaching it.

Moving on from building and visualizing sample spaces, let's learn about their cardinality!

Finite and Infinite Sample Spaces

The examples you've seen so far involve finite sample spaces. A finite sample space is one with a definite number of possible outcomes. Games of chance, like roulette (with numbers from $0$ to $36$) or craps (rolling two $6$-sided dice), most commonly utilize these:

Now, think about an experiment where you toss a coin repeatedly until the first "tails" appears. What will the sample space of this experiment be like? While we expect to stop at some point, there's no way to determine a maximum number of tosses that would assure a "tails". There's always the possibility of turning up another "heads", which makes the related sample space infinite or countably infinite:

Usually, we denote an infinite sample space with three dots—or an ellipsis—that comes after the set's last element. The ellipsis reads as "and so on", indicating that the set continuously extends—an indication of its infinity.

Infinite sample spaces are common, particularly when the outcome is a number lacking an apparent maximum. For instance, in our soccer wagers, we can define a maximum number of red cards, but can we set a limit for the number of goals or fouls? Though, logically, the outcomes of such experiments are finite, there's no way to set a distinct upper limit, so we have to take into account an infinite sample space:

This situation also occurs in online contests. The e-commerce site organizing the contests must estimate participant numbers in a giveaway and calculate the costs. With data like the number of overall purchases, money spent on products, and advertising costs, the statistics team has to deal with infinite sample spaces to make their contest profitable. Questions like—How many will participate in the giveaway? How many will like the site and start using it? How much advertising and prizes should the contest offer—are all things to consider.

Now, you have a solid understanding of sample spaces, but we have yet to talk about events. As they are closely linked to the sample space, let's learn about them next!

Simple and compound events

In Probability, an event is a subset of the sample space. The event can be any occurrence that you are interested in tracking. All the events can be presented as a combination of sample points. Simple events are events that consist of only one outcome in the sample space, and compound events consist of more than one outcome from the sample space.

For example, when rolling a dice there are $6$ possible outcomes and thus the sample space is

In this case, each possible outcome is considered as a simple event. But if we consider an event "rolling an even number" it is a compound event, since it consists of three sample points: $\{2,4,6\}$. This picture illustrates our example.

Before finishing our topic, let's discuss challenges which you may face while working with the sample space.

Sample space challenges

The first challenge which you may face is understanding if the sample space is finite or infinite. Coming back to our examples with soccer, we decided that numbers of goals and fouls do not have upper limits. If we mistakenly thought that those sample spaces are finite, we would make a huge mistake. Such mistakes may influence your further calculations.

The second challenge is understanding the amount of outcomes. Remember our example of tossing two coins? There it was important if the coins were tossed simultaneously of if they were tossed one after another. Thus, there were $3$ or $4$ outcomes respectively. So you should read the problem attentively and make sure not to forget any outcomes when building the sample space! In order to check if your sample space is complete, try to break it by coming up with events and checking if you can pick sample points to reconstruct it. Here illustrations and even simple lists can help you not to lose any outcomes!

You should also remember to ask everything about the task which you are solving in order to understand correctly what should be included into the sample space. Does the order of events matter? How many elements are in the catalog of the e-commerce site? How many players are in each team? Under which condition a game is lost?

Conclusion

Let's summarize what we have learned about the sample space.

• The Sample space, denoted as $\Omega$, is a set of all possible outcomes of an experiment. It can be both finite and infinite.

• The individual outcomes of a sample space, $ω∈Ω$, are called sample points.

• An event is a combination of sample points. It can be both simple and compound.

• It's important to understand the concept of sample space, since probabilities are derived from a well-defined sample space and you will constantly use it while learning probability.

• Building a sample space by writing down all the possible outcomes is helpful only for a subset of a problem.

• The sample space is sensitive to the problem definition.

• Remember that you should test your probability and list the common challenges when constructing a sample space!