Finite Sample Space

This section restricts to sample spaces with at most a finite number of points, and builds the full vocabulary of events on top of that restricted setting.

Events, as subsets of $S$. When $S$ is finite, ANY subset of $S$ counts as an event -- formally, every element of the power set $\mathcal P(S)$ is an event, and an event is simply a collection of sample points. Two extreme events always exist: $S$ itself is the sure event (or certain event -- it is guaranteed to happen, since it contains every possible outcome), and the empty set $\varnothing$ is the impossible event (it can never happen).

Illustration. Take $S=\{1,2,3,4\}$ and list its power set $\mathcal P(S)$: $\varnothing,\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}$ -- all 16 of these are events; $\varnothing$ is impossible; $\{1\},\{2\},\{3\},\{4\}$ are simple/elementary events; and $\{1,2,3,4\}=S$ is the sure event.

Complementary event. For every event $A$, there is a corresponding event $\bar A$ (also written $A'$ or $A^c$) -- the complementary event to $A$, meaning 'not $A$', i.e. $A$ does not occur.

Mutually exclusive events. Events $A_1,A_2,\ldots,A_k$ are mutually exclusive (or disjoint) when they cannot occur simultaneously: $A_i\cap A_j=\varnothing$ for every $i\ne j$. Rolling a die, $\{1,3\}$ and $\{2,4,5,6\}$ are mutually exclusive (they share no outcome); so are $\{1,6\}$ and $\{2,3,5\}$.

Mutually inclusive events. Events $A_1,\ldots,A_k$ are mutually inclusive when they CAN occur simultaneously: $A_i\cap A_j\ne\varnothing$ for some $i\ne j$. On a die, $\{2,3,5\}$ and $\{5,6\}$ are mutually inclusive, since $\{2,3,5\}\cap\{5,6\}=\{5\}\ne\varnothing$.

Exhaustive events. Events $A_1,\ldots,A_k$ are exhaustive when their union covers the whole sample space: $A_1\cup A_2\cup\cdots\cup A_k=S$. On a die, $\{2,3\},\{1,3,5\},\{4,6\}$ together give $\{1,2,3,4,5,6\}=S$, so they are exhaustive; so are $\{2,3\},\{4,6\},\{1,5\}$ -- but $\{1,3,5\},\{4,6\},\{6\},\{1,5\}$ are NOT exhaustive, since their union misses the outcome $2$.

Mutually exclusive AND exhaustive. Events satisfying BOTH conditions together -- $A_i\cap A_j=\varnothing$ for $i\ne j$, and $A_1\cup\cdots\cup A_k=S$ -- form a partition of the sample space. On a die, $\{2,3\},\{4,6\},\{1,5\}$ are pairwise disjoint AND their union is $S$, so they are mutually exclusive and exhaustive together. This partition structure is exactly what Total Probability (Section 12.7) and Bayes' Theorem (Section 12.8) are built on.

Equally likely events. Events with the same chance of occurring are equally likely. Rolling a fair die, every face is equally likely; but a coloured/loaded die (where, say, red comes up more often than the other colours) has faces that are NOT equally likely -- 'equally likely' is a genuine assumption about the physical experiment, not automatic.

Methods to find a sample space. For compound experiments, the sample space is often built as a Cartesian product. Two coins tossed: $S=\{H,T\}\times\{H,T\}=\{HH,HT,TH,TT\}$. A coin tossed and a die rolled together: $S=\{H,T\}\times\{1,\ldots,6\}$, 12 outcomes. In general: …