> [!note] Principle > > $ > P(a_1) = P(a_2) = \cdots = P(a_n) > $ > Let $S$ be a finite [[Sample Space|sample space]] with outcomes ${a_1, a_2, \cdots, a_n}$, such that all of the $a_j$s are physically identical, except for the labels we attach to them. In this case, the [[Probability|probability]] of each outcome is equal unless given evidence to the contrary. > > Suppose that the principle applies to a sample space $S$ where $\#(S) = n$, and let $p$ denote the common value of all $P(a_i)$. Let $\mathcal{P}$ be the [[Partition|partition]] of $S$ into its singleton sets, then > $ > \sum_{i = 1}^{n}p = 1 \quad np = 1 \quad p = \frac{1}{n} > $ > If $A$ is any [[Combination|combination]] of event in $\Sigma(S)$ and $\#(A) = r$, then $P(A) = \frac{r}{n}$.