Properties of a Probability Measure

Topics: Probability Measure - Probability

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(theorem)

Let $\mathbb{P}(A)$ be the probability measure of an event $A$. This probability measure has the following properties:

1. $\mathbb{P}(A^c)=1-\mathbb{P}(A)$
2. $\mathbb{P}(\varnothing )=0$
3. If $A\subset B$, then $\mathbb{P}(B\backslash A)=\mathbb{P}(B)-\mathbb{P}(A)$
4. The probability measure is monotonic. That is:

$$A\subset B⟹\mathbb{P}(A)\le \mathbb{P}(B)$$

Finite Additivity §

(theorem)

Let $A_1,\dots ,A_n\in \mathcal{F}$, where $A_i\cap A_j=\varnothing$ ($i\ne j$; disjoint). Then:

$$\mathbb{P}\left({\cup }_{k=1}^n{A}_k\right)=\sum _{k=1}^n\mathbb{P}(A_k)$$

Finite Subadditivity §

(theorem)

Let $A_1,\dots ,A_n\in \mathcal{F}$ not necessarily disjoint. Then:

$$\mathbb{P}\left({\cup }_{k=1}^n{A}_k\right)\le \sum _{k=1}^n\mathbb{P}(A_k)$$

This property is also known as Boole’s inequality.

Inclusion-Exclusion Rule §

(theorem)

Let $A_1,\dots ,A_n\in \mathcal{F}$ not necessarily disjoint. Then:

$$\begin{array}{rl}\mathbb{P}\left({\cup }_{k=1}^n{A}_k\right)=& \sum _{k=1}^n\mathbb{P}(A_k)-\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n\mathbb{P}(A_i\cap A_j)\\ & +\sum _{\begin{array}{c}i,j,k=1\\ i\ne j\ne k\end{array}}^n\mathbb{P}(A_i\cap A_j\cap A_k)+\dots \\ & +(-1{)}^{n+1}\mathbb{P}(A_i\cap \cdots \cap A_n)\end{array}$$

Few events

Given only two events:

$$\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)$$

Given only three events:

$$\begin{array}{rl}\mathbb{P}(A\cup B\cup C)=& \mathbb{P}(A)+\mathbb{P}(B)+\mathbb{P}(C)\\ & -\mathbb{P}(A\cap B)-\mathbb{P}(B\cap C)-\mathbb{P}(A\cap C)\\ & +\mathbb{P}(A\cap B\cap C)\end{array}$$