From dfc558ca16b4ad115bc2af563b91136970d0b196 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Wed, 21 Jan 2026 09:53:26 +0100
Subject: [PATCH] Add solution for exercise 2c

---
 src/2026-01-30/presentation.tex | 110 ++++++++++++++++++++++++++++----
 1 file changed, 96 insertions(+), 14 deletions(-)

diff --git a/src/2026-01-30/presentation.tex b/src/2026-01-30/presentation.tex
index b7fc815..fd08f7e 100644
--- a/src/2026-01-30/presentation.tex
+++ b/src/2026-01-30/presentation.tex
@@ -351,27 +351,48 @@
     % tex-fmt: on
 
     \vspace*{-5mm}
+    \pause
     \begin{gather*}
         \tilde{S}_N \sim \mathcal{N}(\mu = 150, \sigma^2 = 145{,}5) \\
     \end{gather*}
-    \vspace*{-5mm}
-    \pause\begin{align*}
-        P(\tilde{S}_N \le dx) \ge 0{,}98
-        \hspace*{5mm} &\Rightarrow \hspace*{5mm}
-        P\left(\frac{\tilde{S}_N - E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}
-        \le \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge 0{,}98 \\[3mm]
-        &\Rightarrow \hspace*{5mm}
-        \Phi\left( \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge 0{,}98 \\[3mm]
-        &\Rightarrow \hspace*{5mm}
-        \frac{dx - 150}{\sqrt{145{,}5}} \ge \Phi^{-1}(0{,}98) = 2{,}06 \\[3mm]
-        &\Rightarrow \hspace*{5mm}
-        dx \ge 174{,}8
-    \end{align*}
+    \pause
+    \begin{minipage}[t]{0.2\textwidth}
+        \phantom{placeholder}
+    \end{minipage}
+    \begin{minipage}[t]{0.2\textwidth}
+        \vspace*{-4mm}
+        \centering
+        \begin{gather*}
+            P(\tilde{S}_N \le dx) \ge 0{,}98
+        \end{gather*}
+    \end{minipage}%
+    \pause
+    \begin{minipage}[t]{0.3\textwidth}
+        \vspace*{-15mm}
+        \begin{align*}
+            &\Rightarrow \hspace*{5mm}
+            P\Bigg(\overbrace{\frac{\tilde{S}_N -
+                E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}}^{\sim \mathcal{N}(0,1)}
+            \le \frac{dx - 150}{\sqrt{145{,}5}} \Bigg) \ge 0{,}98
+        \end{align*}
+        \vspace*{-5mm}
+        \pause
+        \begin{align*}
+            &\Rightarrow \hspace*{5mm}
+            \Phi\left( \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge
+            0{,}98 \\[3mm]
+            &\Rightarrow \hspace*{5mm}
+            \frac{dx - 150}{\sqrt{145{,}5}} \ge \Phi^{-1}(0{,}98) =
+            2{,}06 \\[3mm]
+            &\Rightarrow \hspace*{5mm}
+            dx \ge 174{,}8
+        \end{align*}
+    \end{minipage}
+    \pause
 
     \centering
     \vspace*{10mm}
     Der Kessel muss mindestens $175$ Teile fassen
-
 \end{frame}
 
 \begin{frame}
@@ -386,6 +407,67 @@
             Wahrscheinlichkeit von $0{,}98$ nicht überfüllt ist?
     \end{enumerate}
     % tex-fmt: on
+
+    \vspace*{-10mm}
+    \pause
+    \begin{gather*}
+        \tilde{S}_N \sim \mathcal{N}\left(\mu = Np, \sigma^2 = Np(1-p)\right) \\
+    \end{gather*}
+    \pause
+    \begin{minipage}[t]{0.2\textwidth}
+        \vspace*{-4mm}
+        \centering
+        \begin{gather*}
+            P(\tilde{S}_N \le 200) \ge 0{,}98
+        \end{gather*}
+    \end{minipage}%
+    \pause
+    \begin{minipage}[t]{0.3\textwidth}
+        \vspace*{-15mm}
+        \begin{align*}
+            &\Rightarrow \hspace*{5mm}
+            P\Bigg(\overbrace{\frac{\tilde{S}_N -
+                E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}}^{\sim
+                \mathcal{N}(0,1)}
+            \le \frac{200 - Np}{\sqrt{Np(1-p)} } \Bigg) \ge 0{,}98
+        \end{align*}
+        \vspace*{-5mm}
+        \pause
+        \begin{align*}
+            &\Rightarrow \hspace*{5mm}
+            \Phi\left( \frac{200 - Np}{\sqrt{Np(1-p)}} \right) \ge
+            0{,}98 \\[3mm]
+            &\Rightarrow \hspace*{5mm}
+            \frac{200 - Np}{\sqrt{Np(1-p)}} \ge \Phi^{-1}(0{,}98) =
+            2{,}06 \\[3mm]
+            &\Rightarrow \hspace*{5mm}
+            Np + 2{,}06\cdot \sqrt{Np(1-p)} - 200 \le 0
+        \end{align*}
+    \end{minipage}
+    \begin{minipage}[t]{0.4\textwidth}
+        \centering
+        \vspace*{-18mm}
+        \pause
+        \begin{align*}
+            u &:= \sqrt{N} \\
+            a &:= p = 0{,}03 \\
+            b &:= 2{,}06 \cdot \sqrt{p(1-p)} \approx 0{,}351 \\
+            c &:= -200
+        \end{align*}
+        \begin{gather*}
+            \rightarrow \hspace*{5mm} au^2 + bu + c \le 0 \\
+        \end{gather*}
+        \vspace*{-20mm}
+        \pause
+        \begin{align*}
+            \hspace{-17mm}\Rightarrow \hspace{3mm} u = \sqrt{N} \in \left[ -76, 87{,}7 \right] \\
+        \end{align*}
+        \vspace*{-25mm}
+        \pause\begin{align*}
+            &\Rightarrow \hspace{3mm} N \le 5776 \hspace{5mm} \cap \hspace{5mm} N \le 7691{,}29 \\
+            &\Rightarrow \hspace{3mm} N \le 5776
+        \end{align*}
+    \end{minipage}
 \end{frame}
 
 \end{document}