Given an experiment described by the measurable space (Ω,F)\left(\Omega, \mathcal{F}\right) where Ω\Omega is the sample space and F\mathcal{F} is a σ\sigma-algebra on Ω,\Omega, the marginal probability that an event A∈FA\in\mathcal{F} will occur is P(A).\mathbb{P}(A). The probability measure P\mathbb{P} is a function P:F→R\mathbb{P}: \mathcal{F} \rightarrow \mathbb{R} that maps events from F\mathcal{F} to real numbers and satisfies the following properties:

  • For an event A,A, P(A)≥0  ∀ A∈F\mathbb{P}\left(A\right)\geq 0 \,\, \forall \, A\in\mathcal{F}
  • P(∅)=0,\mathbb{P}\left(\varnothing\right)=0, P(Ω)=1\mathbb{P}\left(\Omega\right)=1
  • if E1,E2,…E_1, E_2, \ldots are pairwise disjoint events in F\mathcal{F} then
P(⋃i=1∞Ei)=∑i=1∞P(Ei).\mathbb{P}\left(\bigcup_{i=1}^\infty E_i\right)=\sum_{i=1}^\infty \mathbb{P}\left(E_i\right).

The triple (Ω,F,P)\left(\Omega,\mathcal{F},\mathbb{P}\right) containing a sample space Ω,\Omega, a σ\sigma-algebra F\mathcal{F} on Ω\Omega and a probability measure P\mathbb{P} on (Ω,F)\left(\Omega,\mathcal{F}\right) is a probability space and has the following properties for events A,B∈F:A,B\in\mathcal{F}:

P(A‾)=1−P(A)\mathbb{P}(\overline{A})=1-\mathbb{P}\left(A\right) if  A⊆B  then  P(A)≤P(B)\mathrm{if}\,\,A\subseteq B\,\,\mathrm{then}\,\,\mathbb{P}\left(A\right)\leq\mathbb{P}\left(B\right) P(A∪B)=P(A)+P(B)−P(A∩B)\mathbb{P}\left(A\cup B\right)=\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-\mathbb{P}\left(A\cap B\right) P(A∩B‾)=P(A)−P(A∩B)\mathbb{P}\left(A\cap\overline{B}\right)=\mathbb{P}\left(A\right)-\mathbb{P}\left(A\cap B\right) P(A∪B)≤P(A)+P(B)\mathbb{P}\left(A\cup B\right)\leq\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right) P(A∩B)≥P(A)+P(B)−1.\mathbb{P}\left(A\cap B\right)\geq\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-1.

The joint probability of the finite collection of k>1k>1 events E1,…,Ek∈FE_1,\ldots,E_k\in\mathcal{F} is the probability of the intersection of events E1,…,Ek:E_1,\ldots,E_k:

P(⋂i=1kEi).\mathbb{P}\left(\bigcap_{i=1}^k E_i\right).

Events A,B∈FA,B\in\mathcal{F} are independent events if and only if their joint probability is the product of their marginal probabilities, P(A∩B)=P(A) P(B).\mathbb{P}\left(A\cap B\right)=\mathbb{P}\left(A\right)\,\mathbb{P}\left(B\right).

Disjoint events with non-zero probability of occurring are not independent events. Independence implies that the occurrence of one event does not affect the probability of another event occurring.

A finite collection of k>2k>2 events E1,…,EkE_1, \ldots, E_k are pairwise independent if EiE_i and EjE_j are independent for every pair of events Ei,Ej∈{E1,E2,…,Ek}:E_i, E_j\in \left\{E_1,E_2,\ldots,E_k\right\}:

P(Ei∩Ej)=P(Ei) P(Ej)  ∀ i,j∈{1,2,…,k}, i≠j.\mathbb{P}\left(E_i\cap E_j\right)=\mathbb{P}\left(E_i\right)\,\mathbb{P}\left(E_j\right)\,\,\forall\,i,j\in \left\{1,2,\ldots,k\right\},\,i\neq j.

A finite collection of k>2k>2 events E1,…,EkE_1, \ldots, E_k are mutually independent if, for every subset of events in the collection, the probability of their intersection is the product of their probabilities. For 2≤l≤k2\leq l\leq k and distinct indices 1≤i1<i2<…<il≤k:1\leq i_1<i_2<\ldots<i_l\leq k:

P(⋂j=1lEij)=∏j=1lP(Eij).\mathbb{P}\left(\bigcap_{j=1}^l E_{i_j}\right)=\prod_{j=1}^l\mathbb{P}\left(E_{i_j}\right).

Mutual independence implies pairwise independence, but pairwise independence does not imply mutual independence.