Best way to create colored text boxes

Here a way to define such a box using adjustbox. See the manual for more options.

\documentclass[10pt,a4paper]{article}
\usepackage[hmargin=3cm,vmargin=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[norsk]{babel}
\usepackage[dvipsnames*,svgnames]{xcolor}
\usepackage{mathtools}
\usepackage{xstring}
\usepackage{xparse}
\usepackage{adjustbox}
\usepackage{varwidth}

\newcounter{tittel}
\setcounter{tittel}{0}
\newcounter{problem}
\setcounter{problem}{0}
\newcounter{alternative}
\setcounter{alternative}{0}

\newcounter{navn}[problem]
\renewcommand{\thenavn}{\alph{navn}}
\newcommand{\navn}{\stepcounter{navn}\paragraph*{\thenavn)}}
\setcounter{navn}{0}

\newcommand{\NR}[1]
{ \vspace{5mm} \begin{minipage}[t]{0.051 \textwidth}
 \navn \hspace{0pt}
\end{minipage}
\begin{minipage}[t]{0.949\textwidth}
#1 
\end{minipage}
\vspace{-0.3cm}
}

\newif\iffirstalt

\NewDocumentCommand{\Oppgave}{m o}{%
  \IfNoValueTF{#2}{\setcounter{alternative}{0}\stepcounter{problem}    \firstaltfalse}%
              {\stepcounter{alternative}\iffirstalt\else\stepcounter{problem}    \firstalttrue\fi}
  \section*{Oppgave \arabic{problem}%
             {\normalfont\IfNoValueTF{#2}{}{~Alternative     \Roman{alternative}\ }
                \normalsize (#1 poeng)}%
   \addcontentsline{toc}{section}{Oppgave \arabic{problem} }} 
\vspace{3mm} }

\newenvironment{bluebox}{%
    \noindent
    \adjustbox{innerenv={varwidth}[c]{0.9\linewidth},margin=\fboxsep+.25cm \fboxsep+.2cm,bgcolor=LightSteelBlue,frame,center}\bgroup
}{%
    \egroup
}

\begin{document}

\Oppgave{5}

En gruppe på $8$ elever besår av like mange gutter som jenter. Vi trekker     tilfeldig ut 8 elever.

\NR{ Hva er sannsynligheten for å trekke ut $2$ gutter og $1$ jente? }

\NR{ Hva er sannsynligheten for å trekke ut minst $1$ jente? }


\begin{bluebox}
   Hypergeometrisk     sannsynlighetsfordeling: $\displaystyle P(X=x)=\dfrac{\binom{m}{k}\binom{n-m}{r-k}}{\binom{n}{r}}$ \\
   $m$ elementer i $D$. $n-m$ elementer i $\bar{D}$ \\
   $r$ elementer trekkes tilfeldig. \\
   $X$ er antall elementer som trekkes fra $D$
\end{bluebox}

\end{document}

Here a solution with mdframed. Other possibilities are the packages adjustbox or you simple use tikz (no page breaks available).

The new version is uploaded to CTAN. If you want to test the current example the package can be downloaded here (include also the new documentations)

According to your numbering of paragraphs I would create a new command which does it for you.

\documentclass{article}
\usepackage[hmargin=3cm,vmargin=2cm]{geometry}
\usepackage{amsmath}
\usepackage[dvipsnames*,svgnames]{xcolor}

\usepackage[framemethod=default]{mdframed}

\mdfdefinestyle{NR}{skipabove=\topskip,skipbelow=\topskip,%
                    ,align=center,%
                    innerleftmargin=.25cm,linecolor=black,%
                    linewidth=2pt,backgroundcolor=LightSteelBlue}
\newmdenv[style=NR]{NR}

\newcounter{myparagraph}
\setcounter{myparagraph}{0}
\newcommand\myparagraph[1]{\stepcounter{myparagraph}\paragraph*{\alph{myparagraph})\quad#1}}
\usepackage{showframe}
\begin{document}
\myparagraph{Hva er sannsynligheten for å trekke ut $2$ gutter og $1$ jente?}

\myparagraph{Hva er sannsynligheten for å trekke ut minst $1$ jente?}
\begin{NR}[userdefinedwidth=11cm]
Hypergeometrisk     sannsynlighetsfordeling: $\displaystyle P(X=x)=\dfrac{\binom{m}{k}\binom{n-m}{r-k}}{\binom{n}{r}}$ \\ 
$m$ elementer i $D$. $n-m$ elementer i $\bar{D}$ \\
$r$ elementer trekkes tilfeldig. \\
$X$ er antall elementer som trekkes fra $D$
\end{NR}
\end{document}

A solution with varwidth and tcolorbox, which is there almost out of the box, defining a special bluebox macro:

This declares the \bluebox macro having one optional and one mandatory argument, with blue background, center alignment with respect to page and using at most 0.8\linewidth as width and having no frame.

Other options can be given with the optional argument, see the yellow background for the bluebox ;-)

\newtcbox{bluebox}[1][]{enhanced jigsaw, 
      sharp corners,
      colback=LightSteelBlue,
      center,
      varwidth upper=0.8\linewidth,
      frame hidden,
      #1}

Full code

\documentclass[10pt,a4paper]{article}
\usepackage[hmargin=3cm,vmargin=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[norsk]{babel}
\usepackage[dvipsnames*,svgnames]{xcolor}
\usepackage{mathtools}
\usepackage{varwidth}
\usepackage[most]{tcolorbox}


\newtcbox{bluebox}[1][]{enhanced jigsaw, 
  sharp corners,
  colback=LightSteelBlue,
  center,
  varwidth upper=0.8\linewidth,
  frame hidden,
  #1}

\begin{document}

\section*{Oppgave 2 \normalsize \normalfont (5 poeng)}

En gruppe på $8$ elever besår av like mange gutter som jenter. Vi trekker     tilfeldig ut 8 elever.

\paragraph*{a)} Hva er sannsynligheten for å trekke ut $2$ gutter og $1$ jente? 

\paragraph*{b)} Hva er sannsynligheten for å trekke ut minst $1$ jente? 


\bluebox{%
  Hypergeometrisk sannsynlighetsfordeling: $\displaystyle P(X=x)=\dfrac{\binom{m}{k}\binom{n-m}{r-k}}{\binom{n}{r}}$

 $m$ elementer i $D$. $n-m$ elementer i $\bar{D}$ \\
 $r$ elementer trekkes tilfeldig. \\
 $X$ er antall elementer som trekkes fra $D$  %
}

\bluebox[colback=yellow!60!white]{%
  Hypergeometrisk sannsynlighetsfordeling: $\displaystyle P(X=x)=\dfrac{\binom{m}{k}\binom{n-m}{r-k}}{\binom{n}{r}}$

 $m$ elementer i $D$. $n-m$ elementer i $\bar{D}$ \\
 $r$ elementer trekkes tilfeldig. \\
 $X$ er antall elementer som trekkes fra $D$  %
}



\begin{center}

  \noindent\fcolorbox{black}{LightSteelBlue}{%
    \parbox[t][3cm][c]{11cm}{ \hspace*{0.25cm} Hypergeometrisk     sannsynlighetsfordeling: $\displaystyle P(X=x)=\dfrac{\binom{m}{k}\binom{n-m}{r-k}}{\binom{n}{r}}$ \\ 
    \hspace*{0.25cm} $m$ elementer i $D$. $n-m$ elementer i $\bar{D}$ \\
   \hspace*{0.25cm} $r$ elementer trekkes tilfeldig. \\
   \hspace*{0.25cm} $X$ er antall elementer som trekkes fra $D$  }%
  }%
  \end{center}

\end{document}