Final answer:
The question asks about simplifying a boolean SOP expression for a variable D. Without the explicit boolean expression for D, I cannot provide a detailed simplification process, but standard techniques involve boolean algebra laws and minimalization methods like Karnaugh maps.
Step-by-step explanation:
The detailed analysis you are requesting combines elements from both probability theory and electric circuits, which do not semantically fit together according to the material provided. Therefore, I will focus only on the part of the question that appears to be from the domain of digital logic, specifically boolean algebra or simplifying SOP (Sum of Products) expressions, which fits into the "Computers and Technology" category for a college-level question.
From the context and snippets provided, I have inferred that you are likely asking about simplifying a boolean expression to find the simplest SOP form for a variable D, which is part of a four-variable boolean expression where A is the most significant bit and B is the least significant. Unfortunately, without the explicit boolean expression for D, I cannot provide a step-by-step process for its simplification.
In standard practice, to simplify a boolean expression in SOP form, one would use techniques such as combining common terms, applying boolean algebra laws like the idempotent law, the null law, the domination law, and using Karnaugh maps or the Quine-McCluskey algorithm for minimalization.