probability space

The sample space, \[\Omega\], is the set of outcomes of an experiment (actual real physical outcomes, not numbers or letters, it can even be a very detailed description of the microstate/trajectory of the coin toss). For instance, the sample space of tossing a coin twice would be \[\Omega=\left\{ HH,HT,TT,TH \right\}\] (we write H as a shorthand for heads and t for tails), or for a cubic die \[\Omega=\left\{ 1,2,3,4,5,6 \right\}\] (where each number is the shorthand for how many dots are on a face). Elements of \[\Omega\] are known as outcomes, usually represented as \[\omega\].

The event space, \[\mathcal{F}\] is a sigma-algebra. If \[\Omega\] is countable, it is common for \[\mathcal{F}\] to be assigned the power set of \[\Omega\], i.e. \[\mathcal{F}=\mathcal{P}(\Omega)\]. For a six-sized dice toss, \[\mathcal{F}\] will look like \[\mathcal{F}=\left\{ \emptyset,\left\{ 1 \right\}, \left\{ 2 \right\},\dots,\left\{ 2,3,5 \right\},\dots, \left\{ 1,4,5,6 \right\},\dots,\Omega \right\}\]. Note that every single one of the events in \[\mathcal{F}\] must be associated with a probability (because they must be "measurable"). Naturally, \[\emptyset\] will have a probability of zero (because well, nothing happens), and \[\Omega\] will have a probably of one (because it means if we throw a dice something is guaranteed to happen).

Sets in \[\mathcal{F}\] are known as events. For a two consecutive coin toss, an event might be "at least one head", i.e. \[\left\{ HH,HT,TH \right\}\], or "both coin tosses are the same", i.e. \[\left\{ HH,TT \right\}\].

A probability function \[P\] assigns a number between zero to one or a "probability" to each event in \[\mathcal{F}\]. It is a function, \[P:\mathcal{F}\to \left[ 0,1 \right]\], such that if \[A_{1},A_{2},A_{3},\dots\in \mathcal{F}\] are also mutually disjoint, then \[P \left( \bigcup_{i}A_{i} \right)=\sum_{i=1}P(A_{i})\]. Therefore, by extension the probability of the entire sample space must be equal to one, \[P(\Omega)=1\]. A simple example would be, what is the probability of rolling a one or an even number? The events would be \[\left\{ 1 \right\}\] and \[\left\{ 2,4,6 \right\}\]. Assuming this is a fair dice, the chances of rolling a one would be \[\frac{1}{6}\], and rolling an even would be \[\frac{3}{6}\]. Therefore, the total probability would be \[\frac{4}{6}\]. This, by definition, will be the same as asking "what's the probability of us rolling one of the numbers in the set \[\left\{ 1,2,4,6 \right\}\]", which naturally will also be \[\frac{4}{6}\].