\[ \boxed{p = \frac{110}{29}, \, q = \frac{49}{29}} \]### Question 1

["# Understanding the Probability Expression: ( p = \frac{110}{29}, , q = \frac{49}{29} )", "In probability theory, working with probabilities as exact fractions rather than decimals enhances precision and maintains mathematical rigor. The expression ( \boxed{p = \frac{110}{29}, , q = \frac{49}{29}} ) presents two rational numbers assigned to two events—(p) and (q)—adding up to ( \frac{159}{29} ), which exceeds 1. While probabilities must lie between 0 and 1, this mathematical construct serves as a tool for modeling, teaching, or analytical simplification, especially when exact ratios are preferred over decimal approximations.", "## Context and Interpretation\nAt first glance, ( \frac{110}{29} \approx 3.79 ) and ( \frac{49}{29} \approx 1.69 ), both well outside the valid probability range ([0, 1]). However, such fractions may arise in calculus, conditional probability, or normalization scenarios where scaled probabilities represent relative likelihoods or weights. For example:", "- When working with relative frequencies or scaled distributions, (p) and (q) could represent proportions in a larger framework not yet normalized.\n- They might be components of a partitioned sample space, requiring eventual division by a common factor to yield valid probabilities (e.g., (p = \frac{110}{29}) normalized by ( \frac{159}{29} ) becomes ( \frac{110}{159} )).\n- In Bayesian or probabilistic modeling, such fractions often appear in ratios used for priors, likelihood ratios, or parameter scaling before ensuring total probability sums to 1.", "## Mathematical Insight: Validity and Normalization\nStrictly speaking, neither ( \frac{110}{29} ) nor ( \frac{49}{29} ) is a valid probability, since both exceed 1. For these to represent true probabilities, a normalization step is necessary:\n[\np_{\ ext{norm}} = \frac{110}{110 + 49} = \frac{110}{159}, \quad q_{\ ext{norm}} = \frac{49}{159}\n]\nThis normalization preserves the original ratio while ensuring ( p_{\ ext{norm}} + q_{\ ext{norm}} = 1 ), aligning with the axioms of probability.", "## Applications and Relevance\nWhile the given fractions alone are invalid, their structure reveals key probabilistic concepts:", "- Partitioning Event Spaces: Fractions like these may describe proportional chances within disjoint events, especially in complex systems where denominators reflect total possible outcomes.\n- Model Transformations: In algorithmic probability or simulation, such forms enable preprocessing steps to produce interpretable, valid probabilities through calibration.\n- ** 교육 and Theory: They highlight why direct decimal depiction of probabilities can mislead—precision demands algebraic clarity. For instance, ( p = \frac{110}{29} ) emphasizes numerator dominance, hinting at magnitude even before normalization.", "## Why This Format Matters\nUsing exact fractions avoids cumulative rounding errors and supports symbolic manipulation—crucial in analytical proofs, college-level probability courses, and advanced statistical modeling. For example, conditional probabilities involving (p) or (q) might involve expressions like:\n[\n\Pr(A|B) = \frac{p}{q} \quad \ ext{only if scaled appropriately}\n]\nHere, ( \frac{110/29}{49/29} = \frac{110}{49} ) simplifies cleanly, enabling intuitive likelihood comparisons.", "## Conclusion\nThe expression ( p = \frac{110}{29}, , q = \frac{49}{29} ) serves as a pedagogical and analytical tool, illustrating how exact rational forms underpin probabilistic reasoning—even when normalization is the next logical step. While externalizing these values instantly places them outside valid probability bounds, understanding their structure deepens insight into partitioning, scaling, and transformation within probability theory. For learners and practitioners alike, mastering such representations fosters mathematical precision essential for tackling complex probabilistic models.", "---", "Keywords:** probability fractions, exact probability value, normalized probability, conditional probability framing, mathematical rigor, probability normalization, fractional event probabilities, probabilistic modeling, teaching probability."]