Probability Spaces and Events - Examples and Constructions

Understanding probability spaces through concrete examples illuminates the abstract definitions and reveals how the theory applies to real-world situations.

Discrete Probability Spaces

When the sample space is finite or countably infinite, we have a discrete probability space.

For discrete spaces with countably many outcomes, we need the probability mass function $p: \Omega \to [0,1]$ satisfying $\sum_{\omega \in \Omega} p(\omega) = 1$. Then for any event $A \in \mathcal{F}$: $P(A) = \sum_{\omega \in A} p(\omega)$

Continuous Probability Spaces

For continuous sample spaces like $\Omega = \mathbb{R}$ or $\Omega = [0,1]$, we cannot assign positive probability to individual points.

Product Spaces and Independence

Given two probability spaces $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$, we construct their product space to model independent experiments.

Product Space: $\Omega = \Omega_1 \times \Omega_2$, $\mathcal{F} = \mathcal{F}_1 \otimes \mathcal{F}_2$ (product $\sigma$-algebra), and: $P(A_1 \times A_2) = P_1(A_1) \cdot P_2(A_2)$

Conditional Probability Spaces

Given an event $B$ with $P(B) > 0$, we can construct a new probability space $(\Omega, \mathcal{F}, P_B)$ where $P_B(A) = P(A|B)$.

This "updated" probability space represents our knowledge after learning that $B$ has occurred. All probabilities are now computed relative to the reduced sample space $B$.