Specify where and how edge connects to child (tikz)

You can use options

child anchor=west,
edge from parent macro=\myedgefromparent,

with

\def\myedgefromparent#1#2{
  [style=edge from parent,#1]
  (\tikzparentnode\tikzparentanchor) to #2 (\tikzchildnode\tikzchildanchor)
}

to get

Code:

\documentclass[margin=5pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[
  grow = right,
  child anchor=west,
  edge from parent macro=\myedgefromparent,
  level 1/.style={sibling distance=3em},
  level 2/.style={sibling distance=5em},
  level distance=11em,
  edge from parent/.style={thick,draw=blue!40!black!60},
  punkt/.style = {shape=rectangle, rounded corners,thick,
      draw=blue!40!black!60, align=center,
      top color=white, bottom color=black!20!white,text centered,text width=8em, minimum height = 2em},
  every child node/.append style={punkt,font=\scriptsize},
  edge from parent/.append style={nodes={font=\scriptsize,above,sloped}}
]
\def\myedgefromparent#1#2{
  [style=edge from parent,#1]
  (\tikzparentnode\tikzparentanchor) to #2 (\tikzchildnode\tikzchildanchor)
}
\node[punkt,font=\bfseries] {Question}
child { 
    node {Question} 
    child {
        node {Repetition}
    }
    edge from parent[bend right]
    node {10\%}
}
child { 
    node {Rejection} 
    child {
        node {Agreement}
    }
    edge from parent[bend right]
    node {10\%}
}
child { 
    node {Pass} 
    child {
        node {Acceptance}
    }
    edge from parent
    node {10\%}
}
child { 
    node {Incorrect Answer} 
    child {node[punkt] 
        {Correction by Tutor}
    }
    edge from parent[bend left]
    node {10\%}
}
child { 
    node {Correct Answer} 
    child {
        node {Thanks/Correct}
    }
    edge from parent[bend left]
    node {60\%}
}
;
\end{tikzpicture}
\end{document}

Here's a Forest solution which automatically calculates angles for the edges based on the total number of children and the node's position in the sequence of children.

One major selling point of Forest is that the specification of the tree itself is very concise, once the tree's preamble is configured. If you need the same style of tree several times, you can pull the custom settings out of the preamble and reduce the preamble to a word or two using Forest styles.

\documentclass[tikz,border=10pt,multi]{standalone}
\usepackage{forest}
\begin{document}
\begin{forest}
  punkt/.style = {rounded corners, thick, draw=blue!40!black!60, text centered, top color=white, bottom color=black!20!white, text width=8em, minimum height=2em},
  for tree={%
    grow=0,
    punkt,
    child anchor=parent,
  },
  where level=0{%
    font=\bfseries,
    l sep*=2,
    for children={%
      if={n()==((n_children("!u")+1)/2)}{}{%
        edge path/.wrap pgfmath arg={%
          \noexpand\path [\forestoption{edge}] (!u.parent anchor) [out=#1, in=180] to \forestoption{edge label} (.child anchor);}{90-(180*(n_children("!u")+1-n())/(n_children("!u")+1))},
      },
    },
  }{%
    font=\scriptsize,
  },
  before typesetting nodes={%
    for tree={%
      if edge label={}{}{%
        edge label/.wrap value={node [sloped, midway, above, font=\scriptsize] {#1\%}},
      }
    }
  },
  [Question
    [Question, edge label=10
      [Repetition
      ]
    ]
    [Rejection, edge label=10
      [Agreement
      ]
    ]
    [Pass, edge label=10
      [Acceptance
      ]
    ]
    [Incorrect Answer, edge label=10
      [Correction by Tutor
      ]
    ]
    [Correct Answer, edge label=60
      [Thanks/Correct
      ]
    ]
  ]
\end{forest}
\end{document}

You could use bend left and bend right but it seemed easier to go with out angles for purposes of automating things.