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To understand why the solution manual is so highly sought after, one must first appreciate the difficulty of the subject matter. In early engineering courses, students are taught that inputs produce predictable outputs. If you put a voltage $V$ across a resistor $R$, you get current $I = V/R$. There is no ambiguity.
P(X = 50) = (100 choose 50) * (0.5)^50 * (0.5)^50 ≈ 0.08