From 4e1020dbe06a43d8851560b2f9ffaeff8d3ce985 Mon Sep 17 00:00:00 2001
From: Francesca <111087431+fraktc@users.noreply.github.com>
Date: Sun, 18 Jan 2026 23:32:00 +0100
Subject: [PATCH] Le virgole sono importanti

zao zao, mangia un po' di sushi anche x me :D
---
 .../sections/_language_models.tex                           | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/year2/natural-language-processing/sections/_language_models.tex b/src/year2/natural-language-processing/sections/_language_models.tex
index 1e27d38..1ba9594 100644
--- a/src/year2/natural-language-processing/sections/_language_models.tex
+++ b/src/year2/natural-language-processing/sections/_language_models.tex
@@ -190,7 +190,7 @@
         This is equivalent to computing the probability of the whole sentence, which expanded using the chain rule becomes:
         \[
             \begin{split}
-                \prob{w_1, \dots, w_{i-1} w_i} &= \prob{w_1} \prob{w_2 | w_1} \prob{w_3 | w_{1..2}} \dots \prob{w_n | w_{1..n-1}} \\
+                \prob{w_1, \dots, w_{i-1}, w_i} &= \prob{w_1} \prob{w_2 | w_1} \prob{w_3 | w_{1..2}} \dots \prob{w_n | w_{1..n-1}} \\
                 &= \prod_{i=1}^{n} \prob{w_i | w_{1..i-1}}
             \end{split}
         \]
@@ -224,7 +224,7 @@
                 \begin{description}
                     \item[Estimating $\mathbf{N}$-gram probabilities]
                         Consider the bigram case, the probability that a token $w_i$ follows $w_{i-1}$ can be determined through counting:
-                        \[ \prob{w_i | w_{i-1}} = \frac{\texttt{count}(w_{i-1} w_i)}{\texttt{count}(w_{i-1})} \]
+                        \[ \prob{w_i | w_{i-1}} = \frac{\texttt{count}(w_{i-1}, w_i)}{\texttt{count}(w_{i-1})} \]
                 \end{description}
 
                 \begin{remark}
@@ -378,4 +378,4 @@ Only for $n$-grams that occur enough times a representative probability can be e
                 c^* = \big( \texttt{count}(w_{i-1}w_i) + 1 \big) \frac{\texttt{count}(w_{i-1})}{\texttt{count}(w_{i-1}) + \vert V \vert}
             \]
         \end{example}
-\end{description}
\ No newline at end of file
+\end{description}