Probability Axioms 🧱


## Event
<Definition>
**Event**  

An event is a well-defined set of possible outcomes of the experiment.
</Definition>

For the rolling a dice example, these all can be an event:
- $\{6\}$
- $\{1,6\}$
- $\{1,2,3,4,5,6\}$


<Definition>
**Event**  

An event is a well-defined set of possible outcomes of the experiment.
</Definition>


**Event**  

An event is a well-defined set of possible outcomes of the experiment.


For the rolling a dice example, these all can be an event:
- $\{6\}$
- $\{1,6\}$
- $\{1,2,3,4,5,6\}$



True or false: for an experiment, the sample space is always an event.
<MCQuest totalAttempts="1" correct="A">
* Ture
* False
</MCQuest>


True or false: for an experiment, the sample space is always an event.

<MCQuest totalAttempts="1" correct="A">
* Ture
* False
</MCQuest>



## Probability
<Definition>
**Probability**

A probability, on a sample space $\Omega$ is a specification of numbers $\bold{P}(A)$ for all events $A$ that satisfy Axioms 1, 2, and 3.
</Definition>

$P(A)$ indicates the likelihood that $A$ will occur.

Probabilities are always assigned to events.

Okay, then what are the axioms 1, 2, and 3?


<Definition>
**Probability**

A probability, on a sample space $\Omega$ is a specification of numbers $\bold{P}(A)$ for all events $A$ that satisfy Axioms 1, 2, and 3.
</Definition>


**Probability**

A probability, on a sample space $\Omega$ is a specification of numbers $\bold{P}(A)$ for all events $A$ that satisfy Axioms 1, 2, and 3.


$P(A)$ indicates the likelihood that $A$ will occur.

Probabilities are always assigned to events.

Okay, then what are the axioms 1, 2, and 3?



## Axioms of Probability Theory

<Axiom>
**Axiom 1**

(Non-negativity) $\bold{P}(A)\ge0$ for any event $A$.
</Axiom>

<Axiom>
**Axiom 2**

(Normalization) $\bold{P}(\Omega)=1$, where $\Omega$ is the sample space.
</Axiom>

<Axiom>
**Axiom 3**

(to be strengthened later)  
(Additivity) If $A \cap B=\emptyset$, then $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)$.
</Axiom>

These rules that any probabilistic model should satisfy are called the axioms of probability theory.


<Axiom>
**Axiom 1**

(Non-negativity) $\bold{P}(A)\ge0$ for any event $A$.
</Axiom>


**Axiom 1**

(Non-negativity) $\bold{P}(A)\ge0$ for any event $A$.


<Axiom>
**Axiom 2**

(Normalization) $\bold{P}(\Omega)=1$, where $\Omega$ is the sample space.
</Axiom>


**Axiom 2**

(Normalization) $\bold{P}(\Omega)=1$, where $\Omega$ is the sample space.


<Axiom>
**Axiom 3**

(to be strengthened later)  
(Additivity) If $A \cap B=\emptyset$, then $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)$.
</Axiom>


**Axiom 3**

(to be strengthened later)  
(Additivity) If $A \cap B=\emptyset$, then $\mathrm{P}(A \cup B)=\mathrm{P}(A)+\mathrm{P}(B)$.


These rules that any probabilistic model should satisfy are called the axioms of probability theory.



Let $A$ and $B$ be events on the same sample space, with $\mathbf{P}(A)=0.6$ and $\mathbf{P}(B)=0.7$. Can these two events be disjoint?
<MCQuest totalAttempts="1" correct="B">
* Yes
* No
</MCQuest>


Let $A$ and $B$ be events on the same sample space, with $\mathbf{P}(A)=0.6$ and $\mathbf{P}(B)=0.7$. Can these two events be disjoint?