What is Probability model? — Mathematics

Definition 18.1 High School Mathematics · Chapter 18 — Probability and Random Variables

An experiment with finitely many outcomes is modelled by its sample space $\Omega = \{\omega_1, \dots, \omega_n\}$ (the set of outcomes) together with a probability $\P$ assigning to each outcome $\omega_i$ a number $p_i \geq 0$ , with $p_1 + \dots + p_n = 1$ . An event is a subset $A \subseteq \Omega$ , and

\P(A) = \sum_{\omega_i \in A} p_i .

When all outcomes are equally likely ( $p_i = \frac1n$ : the uniform model),

\P(A) = \frac{\text{number of outcomes in } A} {\text{number of outcomes in } \Omega}.

Examples

Example 18.2

Roll two distinguishable dice: $\Omega$ is the set of the $36$ ordered pairs $(i, j)$ with $1 \leq i, j \leq 6$ , all equally likely. The event $S$ = “the sum is $7$ ” contains the $6$ pairs $(1,6)$ , $(2,5)$ , $(3,4)$ , $(4,3)$ , $(5,2)$ , $(6,1)$ , so $\P(S) = \frac{6}{36} = \frac16$ . The event “the sum is $12$ ” contains only $(6,6)$ : probability $\frac{1}{36}$ . Choosing the right sample space — ordered pairs, not unordered — is what makes the uniform model applicable.

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