Experiments, Sample spaces, and Events

We begin with some basic definitions to relate outcomes of experiments to probabilistic events.

An Experiment is any procedure than can be repeated any number of times (theoretically infinitely), and has a well-defined set of potential Outcomes.

An Outcome, $\omega$, is a potential eventuality of an Experiment.

The Sample Space or 'universe', denoted by $\Omega$, is the set of all possible outcomes $(\omega)$ of the experiment. An outcome is a member of the sample space, so we can write $\omega \in \Omega$.

An Event $A$ is some subset of $\Omega, A \subseteq \Omega$. An elementary event is one which only contains a single outcome.

Example: Sample Space

Whenever a coin is tossed, there are only two possible outcomes: the coin can land heads $(H)$ or tails $(T)$. In other words, the sample space of a coin toss, $\Omega_{\text {coin }}$, can be defined as :

Similarly, the sample space of a common dice roll can result in one of the six distinct outcomes each identified by the number of dots on the top side of the dice:

Sample spaces need not necessarily be finite. For instance, if we were to measure the number of times one needs to roll a dice until the number 6 came up, we could be lucky and roll the six on the first throw of the dice, or very unlucky and never roll it at all (i.e. having to roll an infinite number of times), or somewhere in between (the most likely scenario). This would mean the sample space for this particular experiment, $\Omega_{\mathrm{d} 6}$, contains every positive integer:

where $\mathbb{N}$ is commonly referred to as the set of natural numbers.