Fixed errors.

Author: localhost433

notes/courses/MATH-UA-333/02-axioms.md

@@ -9,24 +9,28 @@ A **sample space** $S$ is the set of all possible outcomes, and an **event** is
 
 ## Probability measure axioms
 
-A probability measure $P$ assigns numbers to events such that $0\le P(E)\le 1$, $P(\emptyset)=0$, $P(S)=1$, and $P(E_1\cup E_2\cup\cdots)=P(E_1)+P(E_2)+\cdots$ for disjoint events.  Probabilities can be interpreted as areas in a Venn diagram.
+A probability measure $P$ assigns numbers to events such that $0\le P(E)\le 1$, $P(\emptyset)=0$, $P(S)=1$, and $P(E_1\cup E_2\cup\cdots)=P(E_1)+P(E_2)+\cdots$ for disjoint events. Probabilities can be interpreted as areas in a Venn diagram.
 
 ### Identities and the inclusion–exclusion principle
 
 From additivity one derives useful identities:
 
 - Complement: $P(A^c)=1-P(A)$.
 - Decomposition: $P(B)=P(A\cap B)+P(B\setminus A)$.
-- Union: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, and more generally for $n$ events the inclusion–exclusion formula sums intersections of increasing size with alternating signs.  The union bound states $P(E_1\cup\cdots\cup E_n)\le P(E_1)+\cdots+P(E_n)$.
+- Union: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, and more generally for $n$ events the inclusion–exclusion formula sums intersections of increasing size with alternating signs. The union bound states $P(E_1\cup\cdots\cup E_n)\le P(E_1)+\cdots+P(E_n)$.
 
 ## Uniform probability on finite spaces
 
-If $S$ has finitely many equally likely outcomes, then $P(E)=\frac{\#E}{\#S}$.
-> Example: two dice have $36$ equally likely outcomes; the event “sum = 7” contains six outcomes, so the probability is $6/36=1/6$.  Drawing 7 cards from a 52‑card deck, the probability of getting a four‑of‑a‑kind is 
+If $S$ has finitely many equally likely outcomes, then
 $$
-P(E)=\frac{13\cdot {48\choose 3}}{{52\choose 7}}.
+P(E)=\frac{\text{Number of elements in } E}{\text{Number of elements in } S}.
 $$
 
+> Example: two dice have $36$ equally likely outcomes; the event “sum = 7” contains six outcomes, so the probability is $6/36=1/6$.  Drawing 7 cards from a 52‑card deck, the probability of getting a four‑of‑a‑kind is 
+> $$
+> P(E)=\frac{13\cdot {48\choose 3}}{{52\choose 7}}.
+> $$
+
 In the birthday problem with 23 people, the probability that at least two share a birthday is $1-\frac{365\cdot364\cdots(365-22)}{365^{23}}$, which exceeds $\tfrac{1}{2}$.
 
 ## Examples and applications

notes/courses/MATH-UA-333/04-05-drv.md

@@ -16,7 +16,7 @@ $$
 > Examples:
 > - Fair die $X\in\{1,\dots,6\}$: $\E[X]=3.5$.
 > - Heads in three fair flips $Y\in\{0,1,2,3\}$: $\E[Y]=1.5$.
-> - Indicator $\mathbf{1}_A$: $\E[\mathbf{1}_A]=P(A)$.
+> - Indicator ${\mathbf{1}}_A$: $\E[{\mathbf{1}}_A]=P(A)$.
 
 ## Linearity of expectation
 For random variables $X,Y$ and scalar $a\in\mathbb{R}$,
@@ -26,7 +26,7 @@ $$
 No independence required.
 
 **Counting-by-indicators trick**
-If $X=\sum_{i=1}^n \mathbf{1}_{A_i}$ counts how many events $A_i$ occur, then
+If $X=\sum_{i=1}^n {\mathbf{1}}_{A_i}$ counts how many events $A_i$ occur, then
 $$
 \E[X] = \sum_{i=1}^n P(A_i).
 $$
@@ -39,15 +39,15 @@ $$
 $$
 Useful identities:
 - For constants $a,b$: $\operatorname{Var}(aX+b)=a^2 \operatorname{Var}(X)$.
-- For an indicator: $\operatorname{Var}(\mathbf{1}_A)=P(A)\big(1-P(A)\big)$.
-- If $X=\sum_{i=1}^n \mathbf{1}_{A_i}$ with independent indicators, then
+- For an indicator: $\operatorname{Var}({\mathbf{1}}_A)=P(A)\big(1-P(A)\big)$.
+- If $X=\sum_{i=1}^n {\mathbf{1}}_{A_i}$ with independent indicators, then
   $$
   \operatorname{Var}(X)=\sum_{i=1}^n P(A_i)\big(1-P(A_i)\big).
   $$
 
 ## Inclusion–Exclusion via indicators
-Using $\mathbf{1}_{A^c} = 1 - \mathbf{1}_A$ and expanding
-$\mathbf{1}_{\cup_i E_i} = 1 - \prod_{i=1}^n (1-\mathbf{1}_{E_i})$, then taking expectations yields
+Using ${\mathbf{1}}_{A^c} = 1 - {\mathbf{1}}_A$ and expanding
+${\mathbf{1}}_{\cup_i E_i} = 1 - \prod_{i=1}^n (1-{\mathbf{1}}_{E_i})$, then taking expectations yields
 $$
 P\Big( \bigcup_{i=1}^n\ E_i \Big)
 = \sum_{k=1}^n (-1)^{k-1}

notes/courses/MATH-UA-333/index.json

@@ -1,6 +1,6 @@
 [
     {
-        "slug": "01-intro",
+        "slug": "01-combinatorics",
         "title": "1 - Basic combinatorics",
         "date": "2025-09-04"
     }

notes/js/course.js

@@ -41,5 +41,5 @@ fetch(`/notes/courses/${slug}/index.json`)
   })
   .catch(e => {
     console.error(e);
-    listEl.innerHTML = "<p>Failed to load notes.</p>";
+    listEl.innerHTML = "<p>Notes are unavailable at the moment, they might not be uploaded yet.</p>";
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