Apparent runaway argument that started with an attempted \begin{align*} environment in \documentclass{amsart}

Question

Here is my document and the problem:

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{float}
\usepackage{graphicx}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
 \graphicspath{ {./images/} }
\begin{document}
\section{The Binomial Distribution}
If $X \sim Bin(n,p)$ where $p = \frac{1}{2}$, hence $q = 1 - \frac{1}{2} = \frac{1}{2}$, then
\begin{align} 
 P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1]{2}}^k \frac{1]{2}^{n-k}\
    &= {n\choose k}{\frac{1]{2}}^n\
    &= \frac{n\choose k}{2^n}
\end{align}
Note that $2^n$ is constant for fixed $n$. Therefore, $P(X=k)$ has symmetry matching the symmetry of Pascal's Triangle.
If $p < \frac{1}{2}$, the distribution is negatively skewed.
\begin{align}
P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1]{6}}^k \frac{5]{6}^{n-k}\
    &= \frac{n\choose k 5^{n-k}}{6^n} 
\end{align}
\end{document}

So there seems to be problems from \section{The Binomial Distribution} onwards. Before when I typeset, it kept saying there was a "missing $ inserted" and I played around with it, unfortunately messing it up so much that now it only has "Undefined control sequence \document class{amsart}" as the error message. Please help!

Follow up question

Here is my entire code (excluding information which I think isn't contributing to an error):

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{float}
\usepackage{graphicx}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
 \graphicspath{ {./images/} }
\begin{document}

$\frac{(n\times p)!}{p!^n}$

\vspace{5mm}

Want to borrow $1$ to $n$ books:\

There are $38$ numbers and only $26$ differences when maximised, so the ceiling function $\lceil \frac{38}{26} \rceil =2$.\
Therefore by PHP, there exists at least two numbers whose difference is at most $26$
\section{Statistics}
\subsection{Averages from grouped data}
\hfill\
\textbf{Estimating the mean:} An estimate for the mean can be obtained by assuming hat each of the raw data values takes the midpoint value of the interval in which it has been placed
$$\text{Mean estimate}  =\frac{\sum{fx}}{\sum{f}}$$
\subsection{Pareto Charts}
\hfill\
The following table shows the 6 most common reasons for conditions failing their driving test.\
\hfill\
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Reason                    & Percentage of conditions & Cumulative frequency \ \midrule
Observations at junctions & 11.9                     & 31                   \
Use of mirrors            & 8.2                      & 52.4                 \
Inappropriate speed       & 5.1                      & 65.7                 \
Steering control          & 4.7                      & 77.9                 \
Reversing around corner   & 4.3                      & 89.1                 \
Incorrect positioning     & 4.2                      & 100                  \ \bottomrule
Total                          &38.4            \
\end{tabular}
\end{center}
\hfill \
\begin{figure}[H]
\centering
\includegraphics[scale=0.15]{paretochart1}
\caption{Pareto chart}
\label{default}
\end{figure}
\newpage
\textbf{Note:} 
\begin{itemize}
\item The Cumulative frequency graph is concave down and monotone increasing
\item Pareto Principle: 80% of consequences come from 20% of the causes
\item Deciles:
    \begin{itemize}
    \item 0 - 10% 1st decile
    \item 10 - 20% 2nd decile
    \item $\vdots$
    \item 90 - 100% 10th decile
    \end{itemize}
\end{itemize}
\section{The Binomial Distribution}
If $X \sim Bin(n,p)$ where $p = \frac{1}{2}$, hence $q = 1 - \frac{1}{2} = \frac{1}{2}$, then
\begin{align} 
 P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1}{2}}^k \frac{1}{2}^{n-k}\
    &= {n\choose k}{\frac{1}{2}}^n\
    &= \frac{{n\choose k}}{2^n}
\end{align}
Note that $2^n$ is constant for fixed $n$. Therefore, $P(X=k)$ has symmetry matching the symmetry of Pascal's Triangle.
If $p < \frac{1}{2}$, the distribution is negatively skewed.
\begin{align}
P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1}{6}}^k \frac{5}{6}^{n-k}\
    &= \frac{{n\choose k 5^{n-k}}}{6^n} 
\end{align}
Note that $5^{n-k}$ does affect the symmetry. If $k$ is small, probabilities to left of the mode are higher. If $k$ is large, probabilities to the right of the mode are lower.
\end{document}

Answer 1

There were two issues: One problem is multiple instances of \frac{1] with mismatched delimiters, and syntax like \frac{n\choose k}{2^n}. Apparently it needs an extra set of braces to unconfuse it.

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{float}
\usepackage{graphicx}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
% \graphicspath{ {./images/} }
\begin{document}
\section{The Binomial Distribution}
If $X \sim Bin(n,p)$ where $p = \frac{1}{2}$, hence $q = 1 - \frac{1}{2} = \frac{1}{2}$, then
\begin{align} 
 P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1}{2}}^k \frac{1}{2}^{n-k}\
    &= {n\choose k}{\frac{1}{2}}^n\
    &= \frac{{n\choose k}}{2^n}
\end{align}
Note that $2^n$ is constant for fixed $n$. Therefore, $P(X=k)$ has symmetry matching the symmetry of Pascal's Triangle.
If $p < \frac{1}{2}$, the distribution is negatively skewed.
\begin{align}
P(X=k) &= {n\choose k}p^k q^{n-k}\
    &= {n\choose k}{\frac{1}{6}}^k \frac{5}{6}^{n-k}\
    &= \frac{{n\choose k 5^{n-k}}}{6^n} 
\end{align}
\end{document}

Answer 2

(This answer addresses the OP's follow-up question, not the question that was posted initially.)

Some general observations about your code:

• In LaTeX documents, one should not use {n \choose k} or {1 \over 2} infix-style notation. Period. Instead, use \binom{n}{k} and \frac{1}{2}, respectively. For more on this subject, see What is the difference between \over and \frac?

• In LaTeX documents, don't use $$ to initiate and terminate display-math mode. Instead, use \[ and \]. For a detailed discussion, please see Why is \[ ... \] preferable to $$ ... $$?

• There's an overabundance of visual formatting directives in your code. For the most part, they don't cause errors, but they're often redundant and create a lot of code clutter. E.g., most of the \hfill\\ directives could be replaced with simple blank lines. Likewise, most of the \\ line terminators in running text can and should be dropped.

• Notation such as {\frac{1}{2}}^k is not conducive to good typography. For one, the curly braces around \frac{1}{2} are redundant. Second, and more importantly, there's very little visual difference between \frac{1}{2}^k and \frac{1^k}{2}. Since the latter impression should obviously be avoided at nearly all cost, please start to write (in a display-math context) \biggl(\frac{1}{2}\biggr)^{\!k}. Your readers will very much appreciate it.

• Don't create paragraph breaks right before display-math material.

• Do please give your readers a (visual) break by replacing diminutive-looking constructs such as \frac{{n\choose k}}{2^n} and \frac{\binom{n}{k 5^{n-k}}}{6^n} with much more reader-friendly alternatives such as \binom{n}{k}2^{-n} and \frac{\binom{n}{k 5^{n-k}}}{6^n}. Try it!

---

Addendum to address the OP's follow-up question, about why I called \frac{{n\choose k}}{2^n} "diminutive looking" and why I suggested using \binom{n}{k}2^{-n} instead. After all, aren't they the same mathematically? Of course they are the same mathematically; however, they're they're not at all the same typographically. TeX and LaTeX are typographic tools, so use them well to maximize the readability of what you're writing.

\documentclass{article}
\usepackage{amsmath} % for \binom macro
\begin{document}
\[
\frac{{n\choose k}}{2^n} \quad\text{vs.}\quad  \binom{n}{k}2^{-n}
\]
\end{document}

Answer 3

If you are using format=pdftex then don't use LaTeX's constructions in your document, i.e. don't use any macros like \documentclass, \usepackage, \begin{document} etc. Use simply plain TeX. It is much more simple than usage of LaTeX.