Tikz isn't plotting the right function

Question

Im trying to plot a probability function in tikz, but the line plotted isn't even close to how the function should look. Im new to using latex and I have no idea why this isnt working, unless tikz just cant handle this function. Here is my code:

\documentclass{article}
\usepackage{pgfplots}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[
    declare function={Probability(\n,\k) = 1-((\n!)/((\n^(\k))*((\n-
\k)!)));}
]

\begin{axis}[
  no markers, domain=0:15, samples=2500,
  axis lines=left, xlabel=$k$, ylabel=$y$,
  every axis y label/.style={at=(current axis.above origin),anchor=east},
  every axis x label/.style={at=(current axis.right of origin),anchor=north},
  height=10cm, width=15cm,
  xtick={0,2,4,6,8,10,12,14},
  ytick={0.1,0.2,0.3,0.4,0.5,.6,.7,.8,.9,1},
  enlargelimits=false, clip=false, axis on top,
  grid = major,
  xmax=15
  ]

\addplot [very thin,cyan!20!black] {Probability(10,x)};


\end{axis}
\end{tikzpicture}
\end{document}

This is what the function should look like, And this is what tikz is plotting

I would appreciate it if someone could point me in the right direction. Thanks

Answer 1

answer addresses only pgfplots aspect. for drawing (vertical) lines you need to add option ycomb:

\documentclass{article}

\usepackage{pgfplots}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[
declare function={p(\n,\k) = 1-((\n!)/((\n^(\k))*((\n-\k)!)));}
                    ]
\begin{axis}[axis lines=middle, 
            xlabel=$k$, ylabel=$y$, 
            ylabel style={anchor=south},
            xlabel style={anchor=west},
            ymax=1.2, xmax=15.5,
            domain=0:15, samples=150,
            no markers
                    ]
\addplot+ [ycomb] {p(2,x)}; % <----
\end{axis}
\end{tikzpicture}

\end{document}

addendum: image with correct settings of your probability function (variables should be integers) is:

\begin{tikzpicture}[
declare function={p(\n,\k) = 1-((\n!)/((\n^(\k))*((\n-\k)!)));}
                    ]
\begin{axis}[axis lines=middle,
            xlabel=$k$, ylabel=$y$,
            ylabel style={anchor=south},
            xlabel style={anchor=west},
            ymax=1.1, xmax=32,      % <-------
            domain=0:30, samples=31,% <-------
%            no markers
                    ]
\addplot+ [ycomb] {p(2,x)};         % <-------
\end{axis}
\end{tikzpicture}