ece108: add probability intro

docs/1b/ece108.md

@@ -846,3 +846,41 @@ $$|A\cup B\cup C|=|A| + |B| + |C| - (|A\cap B| + |A\cap C| + |B\cap C|)-|A\cap B
 If $B$ is a subset of $A$, the dimension of $B$ is related to that of $A$.
 
 $$B\subseteq A\implies|B|=|A|-|\overline B|$$
+
+## Probability
+
+!!! definition
+    - An **experiment** is an event that has a number of outcomes.
+    - **Elementary events** are the outcomes of an experiment compose the set of all events.
+    - An **event** $E$ is a subset of the sample space $S$, which is the **certain event**.
+    - The **null event** is the empty set.
+    - Sets of events are **mutually exclusive** if they are disjoint.
+    - Elementary events are **equiprobable** if they are equally probable.
+    - A **uniform probability distribution** on $S$ is such that all elementary events are equiprobable.
+
+A **probability distribution function (PDF)** $Pr$ converts the elements of the powerset of all outcomes to a real number — its probability.
+
+$$Pr:\mathcal P(S)\to\mathbb R,0\leq P(A)\leq 1$$
+
+A PDF must have, if $S$ is the sample space:
+
+- $\forall A\subseteq S,Pr\{A\}\geq 0$
+- $Pr\{S\}=1$
+- The union of all mutually exclusive sets is the sample space
+
+A **discrete probability distribution** is such that the sample space is a countable set.
+
+For all $A\subseteq S$, the probability of event $A$ is the sum of the probabilities of all elementary events in $A$.
+
+- $Pr\{A\}=\sum_{e\in A}Pr\{\{e\}\}$
+- $Pr\{\empty\}=0$
+- $Pr\{A'\}=1-Pr\{A\}$
+
+Adding events together can never decrease their probability, and the sum of all probabilities must equal $1$ such that $\text{rang}(Pr)\subseteq[0,1]$.
+
+$$A\subseteq B\subseteq S\implies Pr\{A\}\leq Pr\{B\}$$
+
+The **inclusion-exclusion principle** also applies.
+
+$$Pr\{A\cup B\}=Pr\{A\}+Pr\{B\}-Pr\{A\cup B\}$$
+