Random experiments are actions or processes which yield a single outcome from a well-defined set of possible outcomes, known as a sample space, with uncertainty a particular outcome will appear.
A probability space $P$ is a triplet $(\Omega, \mathcal{F}, \Pr)$ with a sample space $\Omega$, event space $\mathcal{F} = \mathcal{P}(\Omega)$, and a probability function $\Pr : \Omega \rightarrow [0, 1]$, which assigns a value from $0$ to $1$ to each $\omega \in \Omega$ such that the probabilities over all $\omega$ sum to 1.
A common example of a probability function is a uniform probability function, appropriate for when any outcome is equally likely, which maps the size of an event $A$ to the proportion of the sample space it contains.
$$ \Pr[A] = \frac{|A|}{|\Omega|} $$
The size of an event $A$ depends on the countability of the sample space. For countable sample spaces, this is the cardinality of $A$. For uncountable sample spaces, it becomes whatever geometrical measure is appropriate to the dimension of the space (length, area, volume, hypervolume).
An event $A \in \mathcal{F}$ is a set of sample points whose probability is simply the sum of the probabilities of its elements. The complement of $A$, denoted $\bar{A} = \Omega \backslash A$, is assigned the probability $A$ does not receive.
$$ \Pr[A] = \sum _{\omega \in A} \Pr [\omega] = 1 - \Pr[\bar{A}] $$
Two events $A$ and $B$ are said to be independent if their respective probabilities do not depend on one another.
$$ \text{$A$ and $B$ are independent} \iff \begin{cases} \Pr[A|B] = A \space \wedge \Pr[B|A] = B \\ \hspace{20pt} \text{or, equivalently, } \\ \Pr[A\cap B] = \Pr[A]\Pr[B]\end{cases} $$
Events $A_1, \space \dots , \space A_n$ are said to be mutually independent if any selection of events, given by indices in $I$, for any $I \sube \{1, \space \dots , \space n\}$, has the following:
$$ \Pr[\bigcap_{i\in I} A_i] = \prod _{i \in I} \Pr[A_i] $$
Events $A_1, \space \dots, \space A_n$ are said to be mutually disjoint if the following holds:
$$ \Pr[\bigcup_{i=1} ^n A_i] = \sum _{i=1} ^n \Pr[A_i] $$