Completed HW0

hw0/hw0.py

@@ -0,0 +1,58 @@
+import collections
+import math
+from typing import List, Set, Tuple
+
+############################################################
+# Custom Type
+# NOTE: You should not modify this.
+
+Position = Tuple[int, int]
+
+
+############################################################
+# Problem 5a
+
+def find_alphabetically_first_word(text: str) -> str:
+    return min(text.split())
+
+
+############################################################
+# Problem 5b
+
+def euclidean_distance(loc1: Position, loc2: Position) -> float:
+    x1, y1 = loc1
+    x2, y2 = loc2
+    return math.sqrt((x2 - x1) ** 2 + (y2 - y1) ** 2)
+
+
+############################################################
+# Problem 5c
+
+def find_non_singleton_words(text: str) -> Set[str]:
+  word_count_lol = collections.Counter(text.split())
+  return {word for word, count in word_count_lol.items() if count > 1}
+
+
+############################################################
+# Problem 5d
+
+def mutate_sentences(sentence: str) -> List[str]:
+    words_in_sentence = sentence.split()
+    similar_sentences = []
+
+    if len(words_in_sentence) < 2:
+      return similar_sentences
+
+    for i in range(len(words_in_sentence) - 1):
+      for j in range(i + 1, len(words_in_sentence)):
+        new_sentence = words_in_sentence[:i] + [words_in_sentence[i], words_in_sentence[j]] + words_in_sentence[j + 1:]
+        similar_sentences.append(' '.join(new_sentence))
+
+    return similar_sentences
+ 
+
+
+if __name__ == "__main__":
+    # feel free to modify this for testing purposes
+    # print(mutate_sentences('the cat and the mouse'))
+   print(find_alphabetically_first_word('hello how are you my apple mango banana'))

main.aux

@@ -3,5 +3,7 @@
 \@writefile{toc}{\contentsline {section}{\numberline {0}Homework 0}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {0.1}About You}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {0.2}Optimization}{2}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\numberline {0.3}Probability}{3}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {0.3}Probability}{4}{}\protected@file@percent }
-\gdef \@abspage@last{5}
+\@writefile{toc}{\contentsline {subsection}{\numberline {0.4}Counting}{6}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {0.5}Programming in Python}{6}{}\protected@file@percent }
+\gdef \@abspage@last{7}

main.fdb_latexmk

@@ -1,10 +1,16 @@
 # Fdb version 4
-["pdflatex"] 1707295945.19537 "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" "main.pdf" "main" 1707295945.95425 0
+["pdflatex"] 1707463862.19733 "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" "main.pdf" "main" 1707463863.15428 0
-  "C:/Users/uzair/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1707288871 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
+  "C:/Users/uzair/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1707457733 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
   "C:/Users/uzair/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1707288887 11548 1854d2b3451e3d517cccd4e0b4daead1 ""
   "C:/Users/uzair/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1707288883 24210786 eea8c83a2618bf979528f763e32e98c9 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1233951854 916 f87d7c45f9c908e672703b83b72241a3 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1233951854 924 9904cf1d39e9767e7a3622f2a125a565 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1233951854 928 2dc8d444221b7a635bb58038579b861a ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1233951854 908 2921f8a10601f252058503cc6570e581 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1233951854 940 75ac932a52f80982a9f8ea75d03a34cf ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1233951854 940 228d6584342e91276bf566bcf9716b83 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmbx10.tfm" 1136765053 1328 c834bbb027764024c09d3d2bf908b5f0 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmbx12.tfm" 1136765053 1324 c910af8c371558dc20f2d7822f66fe64 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr12.tfm" 1136765053 1288 655e228510b4c2a1abe905c368440826 ""
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-  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1247593066 32362 bc3f3eec7ab7d65fe700963d4017d32c ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1247593067 32762 7fee39e011c23b3589931effd97b9702 ""
   "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d ""
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+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/00miktex/epstopdf-sys.cfg" 1616063685 584 2a1075dd71571459f59146da9f7502ad ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsfonts/amsfonts.sty" 1358197772 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsfonts/amssymb.sty" 1358197772 13829 94730e64147574077f8ecfea9bb69af4 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsfonts/umsa.fd" 1358197772 961 6518c6525a34feb5e8250ffa91731cff ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsfonts/umsb.fd" 1358197772 961 d02606146ba5601b5645f987c92e6193 ""
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+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/cancel/cancel.sty" 1388370247 7592 dd751af313a16a0308545d5bfd7aaaa2 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1622999586 13886 d1306dcf79a944f6988e688c1785f9ce ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics-cfg/graphics.cfg" 1465890692 1224 978390e9c2234eab29404bc21b268d1e ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics-def/pdftex.def" 1663915090 19448 1e988b341dda20961a6b931bcde55519 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/graphics.sty" 1665063979 18387 8f900a490197ebaf93c02ae9476d4b09 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/graphicx.sty" 1665063979 8010 a8d949cbdbc5c983593827c9eec252e1 ""
+  "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/keyval.sty" 1665063979 2671 7e67d78d9b88c845599a85b2d41f2e39 ""
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-  "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" 1707295944 5097 898378ef481e2be019a7f5f95119972e ""
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-  "main.aux" 1707295945 464 6dfe8e55ee4a56697c8e62c22d665a53 "pdflatex"
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-  "main.tex" 1707295944 5097 898378ef481e2be019a7f5f95119972e ""
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-  "titlePage.aux" 1707295945 456 c979711f53deacf2b09031977e556db8 "pdflatex"
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main.fls

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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsbsy.sty
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsbsy.sty
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsopn.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\mathtools\mathtools.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\mathtools\mathtools.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\tools\calc.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\tools\calc.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\mathtools\mhsetup.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\mathtools\mhsetup.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\amsfonts.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\amsfonts.sty
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\l3backend\l3backend-pdftex.def
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\l3backend\l3backend-pdftex.def
 INPUT .\main.aux
@@ -26,6 +40,26 @@ INPUT .\titlePage.aux
 INPUT .\titlePage.aux
 INPUT titlePage.aux
 OUTPUT main.aux
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphicx.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphicx.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphics.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphics.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\trig.sty
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+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-cfg\graphics.cfg
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-cfg\graphics.cfg
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty
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+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\00miktex\epstopdf-sys.cfg
 INPUT .\titlePage.tex
 INPUT .\titlePage.tex
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm
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+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm7.tfm
+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm5.tfm
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmti10.tfm
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr12.pfb
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr7.pfb
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb
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 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb
 INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb
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+INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb

main.log

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main.tex

@@ -1,5 +1,9 @@
 \documentclass[titlepage]{article}
 \usepackage{amsmath}
+\usepackage{mathtools}
+\usepackage{cancel}
+\usepackage{amssymb}
+
 
 \setcounter{section}{-1}
 
@@ -8,18 +12,16 @@
 \include{titlePage}
 
 \section*{Introduction}
-This document contains the work I, Uzair Hamed Mohammed, have done as required by this course. This is my first time using \LaTeX \ to format homework, so strange formatting may occur.\\[0.1in]
+This document contains the work I, Uzair Hamed Mohammed, have done as required by this course. This is my first time using \LaTeX \ to type homework, so strange formatting may occur.\\[0.1in]
-The current plan is to append homework assignments to this document as they are assigned. At the end of the semester, the resultant document should be a nice and comprehensive compilation of assignments.\\[0.1in]
+I once read in a math and Linux oriented blog that handwriting your math homework and notes, and then typing it up in \LaTeX \ greatly boosts mastery and memorization of concepts. My plan is to do exactly that, and append homework assignments to this document as they are completed. At the end of the semester, the result should be a nice and comprehensive compilation of assignments.\\[0.1in]
 My current workflow is as follows:
 
 \begin{enumerate}
     \item Read the homework instructions and try to understand the assignment.
     
-    \item Conduct research or referencing to relevant material if needed.
-    
     \item Work by hand, on my tablet, in the Samsung Notes app.
     
-    \item Beautify my work by transcribing it here using LaTeX.
+    \item Beautify my work by transcribing it here using \LaTeX.
 \end{enumerate}
 When it's time to submit an assignment, I will export this document as a PDF file and turn in just the relevant pages. Please let me know what you think of this format!
 
@@ -50,23 +52,50 @@ When it's time to submit an assignment, I will export this document as a PDF fil
     \item [a.] Let a, b, and c be positive real numbers. Consider the quadratic function
         $$f(\theta) = a{\theta}^2 + b\theta + c$$
         Note that $\theta$ here is a real number. What value of $\theta$ minimizes $f(\theta)$?\\[0.1in]
-        Solution:
+        \underline{Solution}:
         Since this is a positive parabola, we can use the formula for the vertex of a parabola, $\frac{-b}{2a}$, to find the minimum:\\
         $$
         \boxed{\theta = \frac{-b}{2a}}
         $$
+
+        \fbox{\parbox{\textwidth}{For the following problems, I'm going to use the general approach of:
+        \begin{enumerate}
+            \item Take derivative of equation
+            \item Set derivative equal to 0
+            \item Solve for $\theta$
+        \end{enumerate}}}
     
     \item [b.] Let $x_1,\dots,x_n$ be real numbers. Consider the quadratic function
         $$g(\theta) = \sum^n_{i=1}(\theta - x_i)^2.$$
-        What value of $\theta$ minimizes $g(\theta)$?
+        What value of $\theta$ minimizes $g(\theta)$?\\
+        \underline{Solution}:
 
-        {\LARGE{TODO}}
+        \begin{align*}
+                             g'(\theta) &= \sum_{i=1}^{n} 2(\theta -x_i)\\
+                                      0 &= \sum_{i=1}^{n} 2(\theta -x_i)\\
+                \theta \sum_{i=1}^{n} 2 &= \sum_{i=1}^{n} 2x_i\\
+                \Aboxed{\theta &= \frac{\sum_{i=1}^{n}x_i}{n}}
+        \end{align*}
+        
 
     \item [c.] Let $x_1,\dots,x_n$ again be real numbers, and let $w_1,\dots,w_n$ be positive real numbers that we can interpret as representing the importance of each of the $x_i$'s. Consider the weighted quadratic function
         $$h(\theta) = \sum_{i=1}^{n}w_i(\theta - x_i)^2.$$
-        What value of $\theta$ minimizes $h(\theta)$?
+        What value of $\theta$ minimizes $h(\theta)$?\\
+        \underline{Solution}:
 
-    \item [d.] What issue could arise in the minimization of $h$ if some of the $w_i$'s are negative?
+        \begin{align*}
+            h'(\theta) &= 2 * \sum_{i=1}^{n} w_i (\theta - x_i)\\
+            0 &= 2 * \sum_{i=1}^{n} w_i (\theta - x_i)\\
+            \frac{0}{2} &= \frac{\cancel{2}*\sum_{i=1}^{n} w_i (\theta - x_i)}{\cancel{2}}\\
+            0 &= \sum_{i=1}^{n} w_i (\theta - x_i)\\
+            0 &= \sum_{i=1}^{n} (w_i \theta - w_i x_i)\\
+            0 &= \sum_{i=1}^{n} w_i \theta - \sum_{i=1}^{n} w_i x_i\\
+            \sum_{i=1}^{n} w_i x_i &= \sum_{i=1}^{n} w_i \theta\\
+            \Aboxed{\frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i } &= \theta}
+        \end{align*}
+
+    \item [d.] What issue could arise in the minimization of $h$ if some of the $w_i$'s are negative?\\
+    \underline{Solution}: If some of the $w_i$'s are negative, it can lead to convoluted results and a possible divergence towards $\infty$
 \end{itemize}
 
 \subsection{Probability}
@@ -76,11 +105,40 @@ When it's time to submit an assignment, I will export this document as a PDF fil
     2-10) in each of the four suits (clubs, diamonds, hearts, spades). If a card is drawn at random, what
     is the probability that it is a spade or a two?
 
+    \underline{Solution}:\\
+    $\rightarrow$ \underline{Let} $P(S)$ be the probability of drawing a spade = $\frac{13}{52} = \frac{1}{4}$\\
+    $\rightarrow$ \underline{Let} $P(T)$ be the probability of drawing a card that is a two = $\frac{4}{52} = \frac{1}{13}$\\
+    
+    $$
+    P(S \cap T) = \frac{1}{52}
+    $$
+
+    The overall probability of drawing a spade or a two is: 
+        \begin{align*}
+            P(S \cup T) &= (P(S) + P(T)) - P(S \cap T)\\
+            &= (\frac{1}{4} + \frac{1}{13}) - \frac{1}{52}\\
+            \Aboxed{&= \frac{4}{13} \approx 30.7\%}
+        \end{align*}
+        
+
     \item [b.] Two factories — Factory A and Factory B — design batteries to be used in mobile phones.
     Factory A produces 60\% of all batteries, and Factory B produces the other 40\%. 2\% of Factory A's
     batteries have defects, and 4\% of Factory B's batteries have defects. What is the probability that a
     battery is both made by Factory A and defective?
 
+    \underline{Solution}:\\
+    $\rightarrow$ \underline{Let} $P(A) = 0.6$ (probability battery is from Factory A)\\
+    $\rightarrow$ \underline{Let} $P(D|A) = 0.02$ (probaility battery is defective if it's from Factory A)\\
+    $\rightarrow$ \underline{Let} $P(A)' = 0.4$ (probability battery is not from Factory A)\\
+    $\rightarrow$ \underline{Let} $P(D|A)' = 0.04$ (probability battery is defective if it's not from Factory A)
+    
+    $\rightarrow$ \underline{Find} $P(A \cap D)$.
+
+    $$\begin{gathered}$$
+        ...\\...\\...\\P(A \cap D) = 0.012 \approx 1.2\%
+    $$\end{gathered}$$
+
+
     \item [c.] Consider the following (made up) facts about COVID incidence and testing:
         \begin{itemize}
             \item In the absence of any special information, the probability that a person has COVID is 1\%.
@@ -91,10 +149,45 @@ When it's time to submit an assignment, I will export this document as a PDF fil
         Suppose you take a COVID test and it reads positive. Given the facts above, what is the probability
 that you have COVID?
 
+        \underline{Solution}:\\
+        $\rightarrow$ \underline{Let} $C$ = COVID\\
+        $\rightarrow$ \underline{Let} $T$ = Positive Test\\[0.5cm]
+        $\rightarrow$ \underline{Let} $P(C) = 0.01$ (probability of having COVID)\\
+        $\rightarrow$ \underline{Let} $P(T|C)$ = $0.8$ (probability of a positive test if a person has COVID)\\
+        $\rightarrow$ \underline{Let} $P(T|C)' = 0.1$ (probability of a false positive)\\
+        $\rightarrow$ \underline{Find} $P(C|T)$ (probability of having COVID with a positive test result)
+
+        \begin{align*}
+            P(T) &= P(T|C) * (P(C) + (P(T|C)' * P(C)'))\\
+            P(T) &= 0.8 * (0.01 + (0.1 * 0.99))\\
+            P(T) &= 0.0872
+        \end{align*}
+
+        \rule{4.4in}{0.5pt}
+
+        \begin{align*}
+            P(C|T) &= \frac{(0.8*0.01)}{0.0872}\\
+                   &= \frac{0.008}{0.0872}\\
+                   \Aboxed{&= \frac{10}{109}\approx 9.17\%}
+        \end{align*}
+
     \item [d. ] Suppose you repeatedly roll a fair six-sided die until you roll a 1 (and then you stop). Every
     time you roll a 3, you win a points, and every time you roll a 6, you lose b points. You do not win or
     lose any points if you roll a 2, 4, or 5. What is the expected number of points (as a function of a and
     b) you will have when you stop?
 \end{itemize}
 
+\subsection{Counting}
+
+\begin{itemize}
+    \item [a.] $n \times n$ grid, rectangle, Big O?\\
+    \underline{Solution}: $O(n^2)$
+
+    \item [b.] Three rectangles, possible ways?\\
+    \underline{Solution}: $n^2 * n^2 * n^2 = n^6$
+\end{itemize}
+
+\subsection{Programming in Python}
+See attached .py file
+
 \end{document}