Before we can talk about the chance of something happening, we first need to list every single thing that could happen. In probability, this complete list of all possible outcomes of an experiment is called the sample space.

Think of an experiment as any action with an uncertain result. Flipping a coin is an experiment. The result is uncertain, but we know all the possibilities. The coin will land on either heads or tails. There are no other options. So, the sample space is simply {Heads, Tails}.

Let's try rolling a standard six-sided die. The possible outcomes are landing on 1, 2, 3, 4, 5, or 6. The sample space, S, is:

What if we make the experiment more complex, like rolling two dice? To find the sample space, we need to list every possible pair of outcomes. The first die can be any number from 1 to 6, and the second die can also be any number from 1 to 6. We can represent each outcome as an ordered pair, like (1, 1) for two ones, or (3, 5) for a 3 on the first die and a 5 on the second.

As you can see, there are 36 possible outcomes when rolling two dice. Listing them all out is crucial. If you miss some, or count some twice, any probability you calculate later will be wrong.

The examples we've seen so far—coin flips, die rolls, card draws—all have finite sample spaces. That means you can count the number of possible outcomes. There's an end to the list, even if it's very long.

But some experiments have an infinite number of outcomes. Imagine an experiment where you flip a coin repeatedly until it lands on heads. What are the possible outcomes? You might get heads on the first flip (H). Or maybe on the second (TH). Or the third (TTH). This could, in theory, go on forever.

The sample space would be $S = \{H, TH, TTH, TTTH, ...\}$. Since this list has no end, it's an infinite sample space.

Clearly defining your experiment is the key. An experiment like "draw two cards from a deck" is too vague. Does the order you draw them in matter? Do you put the first card back before drawing the second? Changing these conditions creates a different experiment with a different sample space.

Getting the sample space right is the foundation. Once you have a complete and accurate list of every possibility, you're ready to start figuring out how likely each of those possibilities is.