An experiment is a process that results in one (and only one) of several possible [[Science/Scientific Method/Measurement|observations]], known as the *outcome* ([[Event|event]]). The [[Set|set]] of all possible outcomes is known as the [[Sample Space|sample space]]. # Principles for Possible Outcomes ### Addition $ n(S) = \sum n_i $ When choosing between $r$ **disjoint** alternatives, if each alternative can produce a number of outcomes $n = \{n_1, n_2, n_3, \cdots\}$, then the total number of possible outcomes is equal to the sum of the outcomes of each alternative. ### Principle of Counting > [!note] Basic Principle of Counting > > Let $A$ and $B$ be two experiences which have $m$ and $n$ outcomes respectively, then the combined experiences $A \circ B$ has $mn$ outcomes. > > *Proof*: Imagine a matrix whose entries represent outcomes of $A \circ B$: > > |(1, 1)|(1, 2)|...|(1, m)| > |--|--|--|--| > |(2, 1)|(2, 2)|...|(2, m)| > |...|...|...|...| > |(n, 1)|(n, 2)|...|(n, m)| > > The size of this matrix is equal to $mn$, representing $mn$ outcomes in total. > [!note] Generalised Principle of Counting > > $n(S) = \prod n_i$ > > Let $A_1$, $A_2$, $A_3$, $...$ be [[Probabilistic Independence|independent]] experiences with $n_1$, $n_2$, $n_3$, $...$ outcomes. The combined experience $A_1 \circ A_2 \circ A_3 \cdots$ has $n_1, n_2, n_3, \cdots$ possible outcomes. > > *Proof by [[Axiom of Induction|induction]]*: The result holds for $r = 2$. If the result holds for $r$, then for $r + 1$, $A_1 \circ A_2 \circ \cdots \circ A_{r} \circ A_{r + 1}$ $=$ $(A_1 \circ A_2 \circ \cdots \circ A_{r}) \circ A_{r + 1}$, the possible number of outcomes is equal to $(n_1n_2 \cdots n_r)n_{r+1}$ by the specific principle, which is $n_1n_2 \cdots n_{r+1}$.