diff --git a/documents/by-course/pstat-120a/course-notes/main.typ b/documents/by-course/pstat-120a/course-notes/main.typ
index efcd39c..ca78488 100644
--- a/documents/by-course/pstat-120a/course-notes/main.typ
+++ b/documents/by-course/pstat-120a/course-notes/main.typ
@@ -10,7 +10,7 @@
 
 #outline()
 
-= Lecture 1
+= Lecture #datetime(day: 6, month: 1, year: 2025).display()
 
 == Preliminaries
 
@@ -349,8 +349,18 @@ This is mostly a formal manipulation to derive the obviously true proposition fr
   This is a stronger result of axiom 3, which generalizes for all sets $A$ and $B$ regardless of whether they're disjoint.
 ]
 
+#remark[
+  These are mostly intuitively true statements (think about the probabilistic
+  concepts represented by the sets) in classical probability that we derive
+  rigorously from our axiomatic probability function $P$.
+]
+
 #example[
+  Now let us consider some trivial concepts in classical probability written in
+  the parlance of combinatorial probability.
+
   Select one card from a deck of 52 cards.
+  Then the following is true:
 
   $
     Omega = {1,2,...,52} \
@@ -374,7 +384,7 @@ This is mostly a formal manipulation to derive the obviously true proposition fr
 == Countable sample spaces
 
 #definition[
-  A sample space $Omega$ is said to be *countable* if its finite or countably infinite.
+  A sample space $Omega$ is said to be *countable* if it's finite or countably infinite.
 ]
 
 In such a case, one can list the elements of $Omega$.