{
  "evaluation_id": "EVAL-20260317-023315",
  "question_id": "EVAL-20260317-023315",
  "question_text": "A disease affects 1 in 10,000 people. A test is 99% sensitive (true positive rate) and 99.5% specific (true negative rate). A patient tests positive. (1) What is the probability they have the disease? (2) If they test positive twice with independent tests, what is the probability? (3) A doctor says 'You tested positive, so you almost certainly have it.' Critique this reasoning. (4) Design a testing protocol that achieves >95% positive predictive value.",
  "category": "reasoning",
  "timestamp": "2026-03-17T02:33:15.571Z",
  "display_date": "Mar 17, 2026",
  "winner": {
    "name": "Qwen 3.5 397B-A17B",
    "provider": "openrouter",
    "score": 10
  },
  "avg_score": 9.796,
  "matrix_size": 24,
  "models_used": [
    {
      "id": "qwen3_coder_next",
      "name": "Qwen 3 Coder Next",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_35b_a3b",
      "name": "Qwen 3.5 35B-A3B",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_27b",
      "name": "Qwen 3.5 27B",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_397b_a17b",
      "name": "Qwen 3.5 397B-A17B",
      "provider": "openrouter"
    },
    {
      "id": "qwen3_8b",
      "name": "Qwen 3 8B",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "qwen35_397b_a17b": {
      "display_name": "Qwen 3.5 397B-A17B",
      "provider": "openrouter",
      "average_score": 10,
      "score_count": 4,
      "min_score": 10,
      "max_score": 10,
      "rank": 1
    },
    "qwen3_8b": {
      "display_name": "Qwen 3 8B",
      "provider": "openrouter",
      "average_score": 9.92,
      "score_count": 5,
      "min_score": 9.8,
      "max_score": 10,
      "rank": 2
    },
    "qwen35_35b_a3b": {
      "display_name": "Qwen 3.5 35B-A3B",
      "provider": "openrouter",
      "average_score": 9.85,
      "score_count": 6,
      "min_score": 9.35,
      "max_score": 10,
      "rank": 3
    },
    "qwen35_27b": {
      "display_name": "Qwen 3.5 27B",
      "provider": "openrouter",
      "average_score": 9.85,
      "score_count": 5,
      "min_score": 9.25,
      "max_score": 10,
      "rank": 4
    },
    "qwen3_coder_next": {
      "display_name": "Qwen 3 Coder Next",
      "provider": "openrouter",
      "average_score": 9.36,
      "score_count": 4,
      "min_score": 8.7,
      "max_score": 10,
      "rank": 5
    }
  },
  "judgments": [
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 9.6,
      "brief_justification": "Accurate calculations for all questions, thorough critique of base rate fallacy, and practical protocol design with mathematical justification. Minor completeness deduction for part (4) not exploring alternative protocols."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response accurately applies Bayes' Theorem, thoroughly explains each part, and provides actionable protocol design. All questions are addressed with mathematical rigor and clear reasoning."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 9.25,
      "brief_justification": "The response correctly applies Bayes' Theorem and addresses all parts of the question. It provides clear explanations and actionable protocol design, though part (4) lacks explicit calculation steps for the three-test scenario."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response accurately applies Bayes' theorem, thoroughly addresses all questions, clearly explains concepts, provides insightful critiques, and offers actionable protocol design with practical implications."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly applies Bayes' Theorem to all four questions, provides clear mathematical derivations, and addresses the base rate fallacy in part (3). The protocol design in part (4) is mathematically rigorous and actionable, though alternative approaches (e.g., targeting higher-risk populations) are not mentioned."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes’ Theorem to all questions, explains the base rate fallacy, and provides actionable solutions for improving PPV. Calculations are precise, and the critique of the doctor’s reasoning is thorough and insightful."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes' Theorem to all four parts, provides clear step-by-step calculations, identifies the base rate fallacy, and proposes a mathematically sound protocol. It is thorough, logically structured, and actionable for real-world scenarios."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes' Theorem to all four questions, provides clear step-by-step calculations, identifies the base rate fallacy in the doctor's reasoning, and proposes a mathematically justified protocol for improving predictive value. It is thorough, logically structured, and offers actionable insights."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response is mathematically rigorous, addresses all four questions comprehensively, explains the base rate fallacy clearly, and proposes a practical protocol with precise calculations. It balances technical accuracy with intuitive explanations and actionable solutions."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes' Theorem, accurately computes PPV for one and two positive tests, clearly explains the base rate fallacy in the doctor's reasoning, and provides a mathematically sound protocol (three positive tests) to achieve >95% PPV, with rigorous derivation and justification."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes' Theorem to all parts, provides accurate numerical calculations, clearly explains the base rate fallacy in part (3), and proposes a mathematically sound three-test protocol achieving >95% PPV in part (4). All reasoning is logically sound, well-structured, and clinically insightful."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes’ Theorem for all parts, provides accurate numerical calculations, clearly explains the base rate fallacy, and proposes a well-justified, mathematically sound testing protocol—verified by computing PPV after three tests (~99.9%) exceeding 95%."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response correctly applies Bayes' Theorem for all parts, provides accurate calculations with intuitive checks, clearly identifies the base rate fallacy in the doctor's reasoning, and proposes a well-justified, mathematically sound testing protocol achieving >95% PPV—demonstrating exceptional thoroughness, clarity, and practical insight."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 10,
      "brief_justification": "The model accurately applied Bayes' theorem to calculate probabilities and correctly identified the base rate fallacy. The derived testing protocol is mathematically sound and directly addresses the requirement for >95% PPV."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 9.15,
      "brief_justification": "The response is mathematically sound and well-structured, effectively explaining the base rate fallacy and designing a valid protocol. It loses points for a minor rounding error in the final percentage (95.1% vs 95.2%) and an abrupt cutoff at the end of the text."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response accurately applies Bayes' Theorem for all calculations, correctly identifies the base rate fallacy, and proposes a mathematically sound protocol to achieve the required PPV. The structure is clear, and the inclusion of real-world context in the protocol section adds significant depth."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 9.8,
      "brief_justification": "The model accurately applies Bayes' Theorem for all calculations and provides a logically sound, mathematically derived protocol design. The response is well-structured, clearly explains the base rate fallacy, and thoroughly addresses every part of the prompt."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 8.7,
      "brief_justification": "The mathematical analysis is accurate and well-structured, but the response cuts off mid-sentence at the end, leaving the final protocol recommendation incomplete."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 9.35,
      "brief_justification": "The mathematical derivations are accurate and the critique of the base rate fallacy is insightful, though Part 2 contains a minor inconsistency between the stated prior and the values used in the calculation. The response is well-structured and provides a practical, mathematically verified protocol."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The response provides mathematically accurate calculations for all parts, clearly explains the base rate fallacy, and proposes a valid testing protocol while acknowledging real-world clinical nuances regarding test independence."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response demonstrates perfect mathematical accuracy and logical validity across all four parts of the problem. It provides exceptional clarity and practical depth, particularly by addressing the assumption of test independence in the protocol design."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 10,
      "brief_justification": "The response is mathematically flawless and clearly structured, correctly identifying the base rate fallacy and deriving the testing protocol."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 9.75,
      "brief_justification": "The response is mathematically rigorous and well-structured, accurately applying Bayes' Theorem and explaining the base rate fallacy. A minor deduction in correctness is due to Part 4 claiming a PPV >99.9% when the calculated value is approximately 99.87%."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The model demonstrates perfect factual accuracy in all Bayesian calculations and correctly identifies the base rate fallacy. The proposed testing protocol is mathematically justified and includes valuable clinical context regarding test independence."
    }
  ],
  "meta": {
    "source": "The Multivac (app.themultivac.com)",
    "methodology": "10x10 blind peer matrix evaluation",
    "criteria": "correctness, completeness, clarity, depth, usefulness",
    "self_judgments": "excluded from rankings",
    "license": "Open data — cite as: The Multivac (2026)"
  }
}