A regular tetrahedron has vertices \(A, B, C,\) and \(D\). Each edge of the tetrahedron has a length of 6 units. A plane passes through the midpoints of edges \(\overline{AB}\), \(\overline{AC}\), and \(\overline{AD}\). What is the probability that a randomly chosen point inside the tetrahedron lies on the same side of the plane as vertex \(D\)? Express your answer as a common fraction. Let's think step by step and output the final answer within \boxed{}.
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