{
  "evaluation_id": "EVAL-20260318-164038",
  "question_id": "EVAL-20260318-164038",
  "question_text": "Here is a flawed solution to a problem. The solution looks correct on the surface but contains a subtle logical error that produces the wrong answer.\n\nProblem: \"A company has 1000 employees. 60% are engineers, 40% are managers. 30% of engineers and 50% of managers speak French. An employee is selected at random and speaks French. What is the probability they are an engineer?\"\n\nFlawed solution:\n\"P(Engineer|French) = P(French|Engineer) * P(Engineer) / P(French)\n= 0.30 * 0.60 / (0.30 + 0.50)\n= 0.18 / 0.80\n= 0.225\n\nSo there is a 22.5% probability the French speaker is an engineer.\"\n\nYour task:\n(1) Find the exact error in the denominator calculation. Show the correct computation step by step.\n(2) Explain WHY this type of error is common. What cognitive shortcut produces it?\n(3) Now apply this to yourself: describe a category of problem where YOU (as an LLM) are most likely to make a similar denominator error. Be specific about the failure mode, not generic.\n(4) Design a self-check protocol (3-5 steps) that you could run after generating any Bayesian calculation to catch this class of error before outputting your answer.",
  "category": "reasoning",
  "timestamp": "2026-03-18T16:40:38.677Z",
  "display_date": "Mar 18, 2026",
  "winner": {
    "name": "GPT-5.4",
    "provider": "openrouter",
    "score": 9.97
  },
  "avg_score": 9.21,
  "matrix_size": 36,
  "models_used": [
    {
      "id": "minimax_m25",
      "name": "MiniMax M2.5",
      "provider": "openrouter"
    },
    {
      "id": "minimax_m1",
      "name": "MiniMax M1",
      "provider": "openrouter"
    },
    {
      "id": "minimax_01",
      "name": "MiniMax-01",
      "provider": "openrouter"
    },
    {
      "id": "judge_claude_sonnet",
      "name": "Claude Sonnet 4.6",
      "provider": "openrouter"
    },
    {
      "id": "judge_gpt54",
      "name": "GPT-5.4",
      "provider": "openrouter"
    },
    {
      "id": "minimax_m27",
      "name": "MiniMax M2.7",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "judge_gpt54": {
      "display_name": "GPT-5.4",
      "provider": "openrouter",
      "average_score": 9.97,
      "score_count": 7,
      "min_score": 9.8,
      "max_score": 10,
      "rank": 1
    },
    "judge_claude_sonnet": {
      "display_name": "Claude Sonnet 4.6",
      "provider": "openrouter",
      "average_score": 9.84,
      "score_count": 7,
      "min_score": 9.2,
      "max_score": 10,
      "rank": 2
    },
    "minimax_m1": {
      "display_name": "MiniMax M1",
      "provider": "openrouter",
      "average_score": 9.67,
      "score_count": 6,
      "min_score": 8.8,
      "max_score": 10,
      "rank": 3
    },
    "minimax_01": {
      "display_name": "MiniMax-01",
      "provider": "openrouter",
      "average_score": 9.31,
      "score_count": 7,
      "min_score": 7.8,
      "max_score": 10,
      "rank": 4
    },
    "minimax_m25": {
      "display_name": "MiniMax M2.5",
      "provider": "openrouter",
      "average_score": 9.05,
      "score_count": 2,
      "min_score": 8.65,
      "max_score": 9.45,
      "rank": 5
    },
    "minimax_m27": {
      "display_name": "MiniMax M2.7",
      "provider": "openrouter",
      "average_score": 7.41,
      "score_count": 7,
      "min_score": 5.3,
      "max_score": 9.6,
      "rank": 6
    }
  },
  "judgments": [
    {
      "judge_key": "minimax_m27",
      "judge_name": "MiniMax M2.7",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.65,
      "brief_justification": "The response correctly identifies the base rate fallacy in the denominator (omitting P(Engineer) and P(Manager) weights), provides accurate step-by-step correction (0.18/0.38 ≈ 47.4%), explains the cognitive mechanism, reflects on LLM failure modes, and designs a systematic self-check protocol. Minor deduction for potential clarity improvements in the self-check protocol."
    },
    {
      "judge_key": "minimax_m27",
      "judge_name": "MiniMax M2.7",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the denominator error (unweighted sum vs. weighted sum of joint probabilities), provides a thorough 4-part answer with excellent depth, uses clear formatting with tables and math notation, and delivers a highly practical self-check protocol with concrete steps."
    },
    {
      "judge_key": "minimax_m27",
      "judge_name": "MiniMax M2.7",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 9.45,
      "brief_justification": "The response correctly identifies the error (using 0.80 instead of 0.38 as the denominator), provides thorough step-by-step calculations, and addresses all four parts of the question with clear structure. Part (3) on LLM failure modes is somewhat generic and could benefit from a more concrete, specific example of a realistic task where this error occurs."
    },
    {
      "judge_key": "minimax_m27",
      "judge_name": "MiniMax M2.7",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the denominator flattening error, provides accurate step-by-step computation yielding 47.4%, insightfully explains the cognitive mechanisms behind it (availability substitution and partition blindness), demonstrates genuine self-awareness with a specific LLM failure mode (multi-stage Bayesian updates in diagnostic testing), and designs a practical 5-step self-check protocol that directly targets this class of error."
    },
    {
      "judge_key": "minimax_m27",
      "judge_name": "MiniMax M2.7",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "This response is exemplary across all dimensions: it correctly identifies the error (summing conditionals instead of weighting joints), provides meticulous step-by-step computation with both symbolic and count-based verification, offers a psychologically insightful explanation of base-rate neglect, candidly admits the LLM's specific failure mode (losing variable-type discipline during pattern-matched generation), and delivers a practical 5-step self-check protocol that addresses the root cause. "
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 7.8,
      "brief_justification": "Parts 1 and 2 are excellent - correct mathematical explanation and good psychological analysis. However, the response is truncated and missing parts 3 (LLM failure mode) and 4 (self-check protocol), making it incomplete."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 10,
      "brief_justification": "The response is excellent: it correctly identifies the denominator error (0.80 vs 0.38), provides step-by-step computation, explains the cognitive shortcut (additive heuristic/simplification bias), acknowledges where LLMs might fail, and provides a practical 5-step self-check protocol. All mathematical explanations are accurate and well-structured."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "Excellent response that correctly identifies the denominator error (failing to weight by population proportions), explains the cognitive mechanisms (availability substitution, partition blindness), provides a specific LLM failure mode (sequential diagnostic testing), and designs a comprehensive 5-step self-check protocol with actionable validation steps."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "The response is exemplary: it correctly identifies the error (adding conditional probabilities instead of weighted joints), provides step-by-step correct computation with verification, insightfully explains the cognitive mechanism (base-rate neglect with type confusion), specifically describes its own LLM failure mode with concrete example, and designs a thorough 5-step self-check protocol. Every component is precise, well-structured, and directly addresses the prompt."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 6.75,
      "brief_justification": "The explanation of the error and cognitive biases is excellent and clear, but the response is incomplete—it cuts off mid-sentence in part (3) and never addresses part (4) at all."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.45,
      "brief_justification": "The response correctly identifies the denominator error (adding probabilities instead of weighted contributions), explains the cognitive bias (base rate neglect), identifies a specific LLM failure mode (over-reliance on pattern matching in probability problems), and provides a practical self-check protocol."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 10,
      "brief_justification": "Response correctly identifies the denominator error (need to weight by base rates), thoroughly explains the base rate fallacy cognitive mechanism, provides a specific LLM failure mode (multi-hop reasoning with intermediate composition changes), and designs a practical 5-step self-check protocol. All mathematical computations are verified and correct."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the denominator error (0.80 vs 0.38), provides step-by-step calculation, explains the additive heuristic cognitive shortcut, identifies a specific LLM failure mode in conditional probability problems, and designs a comprehensive 5-step self-check protocol. All four parts of the question are thoroughly addressed with clear structure and appropriate depth."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "Excellent response: correctly identifies the denominator error (0.38 vs 0.80), provides rigorous explanation of cognitive mechanisms (denominator flattening, availability substitution), identifies a highly specific LLM failure mode (multi-stage diagnostic testing problems), and designs a practical 5-step self-check protocol with clever techniques like the extreme prior sanity test."
    },
    {
      "judge_key": "minimax_m21",
      "judge_name": "MiniMax M2.1",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "Excellent response that correctly identifies the denominator error (summing conditional probabilities instead of joint probabilities), explains the cognitive basis (base-rate neglect), describes a specific LLM failure mode (pattern-matching to shortcut formulas during fluent generation), and provides a comprehensive 5-step self-check protocol. All mathematical computations are accurate and verified with count-based reasoning."
    },
    {
      "judge_key": "minimax_m2",
      "judge_name": "MiniMax M2",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 7.1,
      "brief_justification": "Mathematical explanation is correct and well-structured, with good cognitive analysis, but response is incomplete (cuts off before sections 3-4), significantly limiting practical utility."
    },
    {
      "judge_key": "minimax_m2",
      "judge_name": "MiniMax M2",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 10,
      "brief_justification": "Response correctly identifies the denominator error (forgetting to weight by base rates), explains the base rate fallacy mechanism, provides a specific LLM failure mode (averages of averages), and designs a practical 5-step self-check protocol. All four required parts are addressed with precise mathematical notation and clear explanations."
    },
    {
      "judge_key": "minimax_m2",
      "judge_name": "MiniMax M2",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the denominator error (0.80 vs 0.38), provides step-by-step correct computation, explains the cognitive mechanism (additive heuristic/simplification bias), offers genuine self-reflection on LLM failure modes in conditional probability problems, and designs a comprehensive 5-step self-check protocol. All four parts of the question are thoroughly addressed with precise mathematical notation and clear explanations."
    },
    {
      "judge_key": "minimax_m2",
      "judge_name": "MiniMax M2",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "Response correctly identifies the denominator error (unweighted sum of conditional probabilities), explains the cognitive mechanisms (availability substitution and partition blindness), provides a specific LLM failure mode (multi-stage diagnostic testing problems), and designs a comprehensive 5-step self-check protocol with practical checks including weight verification, prior unity checks, and extreme prior sanity tests."
    },
    {
      "judge_key": "minimax_m2",
      "judge_name": "MiniMax M2",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "Response provides mathematically correct solution with clear step-by-step computation, thoroughly explains the cognitive basis for the error (base-rate neglect, conditional vs. joint probability confusion), identifies a specific LLM failure mode in multi-branch Bayesian problems, and designs a comprehensive 5-step self-check protocol covering type-labeling, law of total probability application, count conversion, plausibility bounds, and joint-probability verification."
    },
    {
      "judge_key": "minimax_m1",
      "judge_name": "MiniMax M1",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 7.25,
      "brief_justification": "The core mathematical error identification and correction are perfect (10), and the explanation of cognitive biases is strong (8). However, the response is severely incomplete: it cuts off mid-sentence, failing to address parts (3) and (4) of the question entirely. The answer stops abruptly at 'Prevalence of disease' without completing the failure mode description or providing any self-check protocol."
    },
    {
      "judge_key": "minimax_m1",
      "judge_name": "MiniMax M1",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the denominator error (confusing conditional probabilities P(French|Engineer) + P(French|Manager) with joint probabilities), provides detailed calculations showing P(French) = 0.38, and arrives at the correct ~47.37% answer. It thoroughly addresses all four parts of the question with clear structure, explains the additive heuristic/simplification bias as the cognitive cause, identifies specific LLM failure modes in conditional probability problems, and provides "
    },
    {
      "judge_key": "minimax_m1",
      "judge_name": "MiniMax M1",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "The response is exemplary in every dimension. It correctly identifies the denominator error (0.80 vs 0.38), provides the accurate final probability (~47.4%), explains the cognitive mechanisms (denominator flattening, availability substitution, partition blindness), identifies a highly specific failure mode for LLMs (multi-stage diagnostic testing), and provides a comprehensive 5-step self-check protocol with concrete, actionable checks like weight verification and extreme prior sanity testing."
    },
    {
      "judge_key": "minimax_m1",
      "judge_name": "MiniMax M1",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "Excellent response addressing all four parts correctly. Identifies the exact error (confusing conditional probabilities with joint probabilities in the denominator), explains the cognitive basis (base-rate neglect), describes a specific LLM failure mode (pattern-matching to shortcuts when conditional probabilities are presented symmetrically), and provides a comprehensive 5-step self-check protocol. Mathematically correct with proper verification via both formula and count-based methods."
    },
    {
      "judge_key": "minimax_01",
      "judge_name": "MiniMax-01",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 9.6,
      "brief_justification": "The response correctly identifies the error in the denominator, explains why such errors are common with detailed cognitive insights, and provides a specific category of failure modes for LLMs along with a self-check protocol."
    },
    {
      "judge_key": "minimax_01",
      "judge_name": "MiniMax-01",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the error, explains the base rate fallacy, provides a specific and relevant example of the AI's potential failure mode, and offers a comprehensive self-check protocol for Bayesian calculations."
    },
    {
      "judge_key": "minimax_01",
      "judge_name": "MiniMax-01",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.65,
      "brief_justification": "The response demonstrates a comprehensive understanding of the problem, correctly identifies the error, explains the cognitive mechanisms behind it, and provides a detailed self-check protocol. It misses a perfect score in depth and usefulness only because it could have further elaborated on additional real-world examples or limitations of the self-check."
    },
    {
      "judge_key": "minimax_01",
      "judge_name": "MiniMax-01",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually accurate, thoroughly explains the error, and provides a clear step-by-step solution. It offers a detailed analysis of why such errors occur and proposes a practical self-check protocol, though it could further explore broader implications of cognitive biases in probabilistic reasoning."
    },
    {
      "judge_key": "judge_claude_sonnet",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 8.1,
      "brief_justification": "The response correctly identifies the denominator error, provides accurate step-by-step computation (47.4%), and gives a strong cognitive explanation. However, the response appears truncated—parts 3 and 4 are incomplete or cut off, leaving the self-check protocol and the specific LLM failure mode unfinished, which significantly reduces completeness and usefulness scores."
    },
    {
      "judge_key": "judge_claude_sonnet",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 9.65,
      "brief_justification": "The response correctly identifies the exact error (unweighted sum of conditional probabilities), provides the correct calculation (47.4%), offers a cognitively insightful explanation of base rate neglect, gives a specific and honest self-assessment of LLM failure modes, and designs a practical 5-step self-check protocol. The depth is excellent throughout, with the only minor gap being that Step 4's sanity check (answer between the two conditionals) isn't universally true and could use a caveat."
    },
    {
      "judge_key": "judge_claude_sonnet",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 8.1,
      "brief_justification": "The response correctly identifies and fixes the denominator error with clear step-by-step work, arriving at the right answer of ~47.37%. However, parts (3) and (4) are somewhat generic—the self-reflection on LLM-specific failure modes lacks concrete specificity (e.g., naming particular problem types like multi-step word problems or nested conditionals where pattern-matching to surface structure causes the error), and the self-check protocol reads as reasonable but abstract rather than operationa"
    },
    {
      "judge_key": "judge_claude_sonnet",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "judge_gpt54",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "The response is exemplary across all dimensions: it precisely identifies the type-mismatch error (conditional vs. joint probabilities in the denominator), provides both algebraic and count-based verification, offers a cognitively insightful explanation of base-rate neglect, gives a genuinely specific and self-aware account of the LLM failure mode (variable-type discipline loss during fluent generation), and delivers a rigorous, actionable 5-step self-check protocol. No meaningful gaps or errors."
    },
    {
      "judge_key": "judge_gpt54",
      "judge_name": "GPT-5.4",
      "respondent_key": "minimax_m27",
      "respondent_name": "MiniMax M2.7",
      "weighted_score": 5.3,
      "brief_justification": "The response correctly identifies and fixes the denominator error and gives a reasonable explanation of the underlying heuristic, but it is truncated during part (3) and does not complete parts (3) and (4). Clear where present, but incomplete overall for the full task."
    },
    {
      "judge_key": "judge_gpt54",
      "judge_name": "GPT-5.4",
      "respondent_key": "minimax_m1",
      "respondent_name": "MiniMax M1",
      "weighted_score": 8.8,
      "brief_justification": "The response correctly identifies the denominator error, computes the right posterior, and gives a solid explanation plus a practical self-check protocol. The only notable issue is Step 4 of the protocol, which states an invalid general sanity check: P(A|X) does not in general have to lie between P(X|A) and P(X|B)."
    },
    {
      "judge_key": "judge_gpt54",
      "judge_name": "GPT-5.4",
      "respondent_key": "minimax_01",
      "respondent_name": "MiniMax-01",
      "weighted_score": 7.8,
      "brief_justification": "The response correctly identifies and fixes the denominator error, computes the right posterior, and provides a reasonable self-check protocol. It loses points because the explanation of why the error is common is somewhat imprecise, and the LLM-specific failure mode remains a bit generic rather than sharply specifying denominator-construction errors in Bayesian mixtures."
    },
    {
      "judge_key": "judge_gpt54",
      "judge_name": "GPT-5.4",
      "respondent_key": "judge_claude_sonnet",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.2,
      "brief_justification": "The response correctly identifies and fixes the denominator error, gives a strong explanation of the cognitive shortcut behind it, provides a specific LLM failure mode, and proposes a practical self-check protocol. Minor nitpick: the claim that the denominator 'must be between 0.30 and 0.50' is context-dependent and slightly overstated, though valid here."
    }
  ],
  "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)"
  }
}