The highest of three dice is a 5: every AI model's reply, tested

You roll three fair six-sided dice. What's the probability that the highest number showing is exactly 5? Give it as a fraction in lowest terms.
Show your working briefly, then end with a line that says "Final answer: " followed by the answer alone.

Exactly what this prompt's replies are checked against, with every other prompt of our test runs.

1. GPT-6 Luna

   Passed: Final answer 61/216: right.

   There are \(5^3\) outcomes where every die is at most 5, and \(4^3\) where every die is at most 4. Thus the probability is
   \[
   \frac{5^3-4^3}{6^3}=\frac{61}{216}.
   \]
   
   Final answer: 61/216

   393 tokens in, 117 out (39 of them reasoning) · 2.2 s · $0.0001 · 1 message on Pro · answered by openai/gpt-6-luna via OpenAI · September 27, 2026

2. GLM 5.3 Flash

   Passed: Final answer 61/216: right.

   P(max ≤ 5) = (5/6)³ = 125/216, P(max ≤ 4) = (4/6)³ = 64/216.
   
   So P(max = 5) = 125/216 − 64/216 = 61/216.
   
   Final answer: 61/216

   409 tokens in, 97 out (27 of them reasoning) · 0.9 s · $0.0001 · 1 message on Pro · answered by z-ai/glm-5.3-flash via Together · September 27, 2026

3. DeepSeek V4.1 Flash

   Passed: Final answer 61/216: right.

   Total outcomes: 6³ = 216.
   
   Max exactly 5 means all dice show ≤ 5, but not all show ≤ 4:
   
   - All ≤ 5: 5³ = 125
   - All ≤ 4: 4³ = 64
   
   So favorable outcomes: 125 − 64 = 61.
   
   Probability = 61/216 (already in lowest terms, since 61 is prime and doesn't divide 216).
   
   Final answer: 61/216

   432 tokens in, 155 out (55 of them reasoning) · 2.5 s · $0.0002 · 1 message on Pro · answered by deepseek/deepseek-v4.1-flash via CoreWeave · September 27, 2026

4. GLM 5.3

   Passed: Final answer 61/216: right.

   All outcomes ≤5 minus all ≤4, divided by 216:
   
   (5³ − 4³)/216 = (125 − 64)/216 = 61/216
   
   61 is prime, so this is already lowest terms.
   
   Final answer: 61/216

   432 tokens in, 79 out (23 of them reasoning) · 0.9 s · $0.0006 · 1 message on Pro · answered by z-ai/glm-5.3 via Wafer · September 27, 2026

5. DeepSeek V4 Pro

   Passed: Final answer 61/216: right.

   The highest die is exactly 5 means:
   
   - All three dice show 5 or less: \(5^3 = 125\) outcomes
   - Subtract outcomes where all dice show 4 or less: \(4^3 = 64\)
   
   So favorable outcomes:
   
   \[
   125 - 64 = 61
   \]
   
   Total outcomes:
   
   \[
   6^3 = 216
   \]
   
   Probability:
   
   \[
   \frac{61}{216}
   \]
   
   Final answer: 61/216

   405 tokens in, 213 out (117 of them reasoning) · 3.5 s · $0.0006 · 1 message on Pro · answered by deepseek/deepseek-v4-pro-0813 via Alibaba · September 27, 2026

6. Gemini 3.8 Flash

   Passed: Final answer 61/216: right.

   To find the probability that the highest number showing on three fair six-sided dice is exactly 5:
   
   1. The total number of possible outcomes when rolling three dice is $6^3 = 216$.
   2. The number of outcomes where all three dice show a number less than or equal to 5 is $5^3 = 125$.
   3. The number of outcomes where all three dice show a number less than or equal to 4 (meaning no 5 is rolled) is $4^3 = 64$.
   4. The number of outcomes where the maximum number is exactly 5 is $125 - 64 = 61$.
   
   Thus, the probability is $\frac{61}{216}$.
   
   Final answer: 61/216

   421 tokens in, 392 out (220 of them reasoning) · 4.9 s · $0.0018 · 1 message on Pro · answered by google/gemini-3.8-flash via Google · September 27, 2026

7. Claude Haiku 4.5

   Passed: Final answer 61/216: right.

   I need to find the probability that the maximum of three dice is exactly 5.
   
   **Working:**
   
   For the maximum to be exactly 5:
   - At least one die shows 5
   - No die shows 6
   
   **Count favorable outcomes:**
   - All three dice must show values from {1, 2, 3, 4, 5}
   - At least one die must show 5
   
   Using complementary counting:
   - Total ways to roll three dice with values in {1,2,3,4,5}: 5³ = 125
   - Ways with no 5 (all from {1,2,3,4}): 4³ = 64
   - Ways with at least one 5: 125 - 64 = 61
   
   **Total possible outcomes:** 6³ = 216
   
   **Probability:** 61/216
   
   Let me verify this is in lowest terms by checking if 61 and 216 share factors:
   - 61 is prime
   - 216 = 2³ × 3³
   - Since 61 is prime and doesn't equal 2 or 3, gcd(61, 216) = 1
   
   Final answer: 61/216

   420 tokens in, 276 out · 2.2 s · $0.0018 · 1 message on Pro · answered by anthropic/claude-haiku-4.5 via Amazon Bedrock · September 27, 2026

8. GPT-6 Sol

   Passed: Final answer 61/216: right.

   For the highest number to be exactly 5, all three dice must show 5 or less, but not all can show 4 or less.
   
   \[
   P(\text{highest is }5)=\left(\frac56\right)^3-\left(\frac46\right)^3
   =\frac{125-64}{216}=\frac{61}{216}.
   \]
   
   Final answer: 61/216

   393 tokens in, 122 out (31 of them reasoning) · 2.2 s · $0.0020 · 1 message on Pro · answered by openai/gpt-6-sol via OpenAI · September 27, 2026

9. Claude Sonnet 5.5

   Passed: Final answer 61/216: right.

   P(max ≤ 5) = (5/6)³ = 125/216
   P(max ≤ 4) = (4/6)³ = 64/216
   
   P(max = 5) = 125/216 − 64/216 = 61/216
   
   61 is prime and doesn't divide 216, so the fraction is already in lowest terms.
   
   Final answer: 61/216

   541 tokens in, 114 out · 1.7 s · $0.0022 · 1 message on Pro · answered by anthropic/claude-sonnet-5.5 via Anthropic · September 28, 2026

10. Kimi K3

    Passed: Final answer 61/216: right.

    Count outcomes where all dice are at most 5: \(5^3=125\).  
    Subtract outcomes where all dice are at most 4: \(4^3=64\).  
    So exactly highest 5: \(125-64=61\) out of \(6^3=216\).
    
    Final answer: 61/216

    480 tokens in, 176 out (96 of them reasoning) · 2.4 s · $0.0027 · 1 message on Pro · answered by moonshotai/kimi-k3 via Phala · September 27, 2026

11. Claude Sonnet 5

    Passed: Final answer 61/216: right.

    P(max ≤ 5) = (5/6)³ = 125/216
    P(max ≤ 4) = (4/6)³ = 64/216
    
    P(max = 5) = 125/216 − 64/216 = 61/216
    
    61 is prime and doesn't divide 216, so this is already in lowest terms.
    
    Final answer: 61/216

    537 tokens in, 174 out (63 of them reasoning) · 3.9 s · $0.0028 · 1 message on Pro · answered by anthropic/claude-sonnet-5 via Claude Platform on AWS · September 27, 2026

12. Grok 4.7

    Passed: Final answer 61/216: right.

    The total number of outcomes when rolling three fair six-sided dice is \(6^3 = 216\).
    
    The highest face is exactly 5 precisely when every die shows a face in \(\{1,2,3,4,5\}\) and at least one die shows a 5. There are \(5^3 = 125\) outcomes in which every face is at most 5, and \(4^3 = 64\) outcomes in which every face is at most 4. Subtracting gives
    \[
    125 - 64 = 61
    \]
    favorable outcomes.
    
    Thus the desired probability is \(\frac{61}{216}\). (The fraction is already in lowest terms, since 61 is prime and does not divide 216.)
    
    Final answer: 61/216

    1,628 tokens in, 786 out (621 of them reasoning) · 9.1 s · $0.0050 · 1 message on Pro · answered by x-ai/grok-4.7 via xAI · September 27, 2026

13. Claude Opus 5.5

    Passed: Final answer 61/216: right.

    P(max = 5) = P(all ≤ 5) − P(all ≤ 4)
    
    - P(all ≤ 5) = (5/6)³ = 125/216
    - P(all ≤ 4) = (4/6)³ = 64/216
    
    So P(max = 5) = (125 − 64)/216 = 61/216.
    
    61 is prime and doesn't divide 216, so this is already in lowest terms.
    
    Final answer: 61/216

    539 tokens in, 178 out (34 of them reasoning) · 3.9 s · $0.0057 · 1 message on Pro · answered by anthropic/claude-opus-5.5 via Claude Platform on AWS · September 27, 2026

14. GPT-6 Astra

    Passed: Final answer 61/216: right.

    There are \(6^3=216\) equally likely outcomes. For the highest number to be exactly 5, all dice must show at most 5, but not all at most 4.
    
    Thus,
    \[
    P(\text{highest}=5)=\frac{5^3-4^3}{6^3}
    =\frac{125-64}{216}
    =\frac{61}{216}.
    \]
    
    Final answer: 61/216

    393 tokens in, 97 out · 2.6 s · $0.0088 · 1 message on Pro · answered by openai/gpt-6-astra via OpenAI · September 27, 2026

15. Gemini 3.1 Pro

    Passed: Final answer 61/216: right.

    To find the probability that the highest number showing on three rolled dice is exactly 5, we can use the difference between two probabilities:
    1. The probability that all three dice show a 5 or lower.
    2. The probability that all three dice show a 4 or lower.
    
    The number of ways all three dice can show 5 or lower is $5^3 = 125$.
    The number of ways all three dice can show 4 or lower is $4^3 = 64$.
    
    The number of outcomes where the highest number is exactly 5 is the difference: $125 - 64 = 61$.
    
    The total number of possible outcomes when rolling three six-sided dice is $6^3 = 216$.
    
    Therefore, the probability is 61/216. Since 61 is a prime number and doesn't divide 216, this fraction is in lowest terms.
    
    Final answer: 61/216

    421 tokens in, 676 out (463 of them reasoning) · 8.3 s · $0.0090 · 1 message on Pro · answered by google/gemini-3.1-pro-preview via Google · September 27, 2026

16. Claude Fable 5.1

    Passed: Final answer 61/216: right.

    P(max ≤ 5) = (5/6)³ = 125/216
    P(max ≤ 4) = (4/6)³ = 64/216
    P(max = 5) = 125/216 − 64/216 = 61/216
    
    61 is prime, so the fraction is already in lowest terms.
    
    Final answer: 61/216

    539 tokens in, 105 out · 5.9 s · $0.0106 · 1 message on Pro · answered by anthropic/claude-fable-5.1 via Anthropic · September 27, 2026

• Pens at 3 for $4
• A discount, then sales tax
• Compound interest over three years
• Four-digit numbers whose digits sum to 9
• The best AI for math