Two dice that add up to 8: every AI model's reply, tested

Solve this math problem step by step. Number the steps, check the result, and end with a line that starts "Final answer:".

---
Two fair six-sided dice are rolled. What is the probability that the numbers add up to 8?

How the AI math solver's test inputs are scored, with every model's results on all five.

1. GPT-6 Luna

   Passed: Final answer 5/36: right.

   1. Each die has 6 possible outcomes, so there are 6 × 6 = 36 equally likely ordered outcomes.
   
   2. The outcomes that add up to 8 are (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
   
   3. There are 5 favorable outcomes, so the probability is 5/36.
   
   4. Check: the possible pairs listed each sum to 8, and there are no others.
   
   Final answer: 5/36

   347 tokens in, 172 out (53 of them reasoning) · 2.5 s · $0.0001 · 1 message on Pro · answered by openai/gpt-6-luna via OpenAI · September 28, 2026

2. GLM 5.3 Flash

   Passed: Final answer 5/36 (about 0.1389): right.

   Step 1: Count total outcomes.
   Two dice, each with 6 faces: 6 × 6 = 36 equally likely outcomes.
   
   Step 2: Count outcomes where the sum is 8.
   List the pairs (first die, second die):
   (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
   That is 5 favorable outcomes.
   
   Step 3: Compute the probability.
   P(sum = 8) = 5/36
   
   Step 4: Check.
   The favorable pairs each sum to 8: 2+6=8, 3+5=8, 4+4=8, 5+3=8, 6+2=8. A count of 5 out of 36 matches the known distribution of sums for two dice (sums of 8 occur 5 times).
   
   Final answer: 5/36 (about 0.1389)

   355 tokens in, 207 out (2 of them reasoning) · 3.1 s · $0.0002 · 1 message on Pro · answered by z-ai/glm-5.3-flash via Friendli · September 28, 2026

3. Claude Haiku 5.5

   Passed: Final answer 5/36: right.

   Step 1. Count the total number of equally likely outcomes.
   Each die has 6 faces, so the number of ordered outcomes is 6 x 6 = 36.
   
   Step 2. List the outcomes whose sum is 8.
   The pairs (first die, second die) with sum 8 are:
   (2,6), (3,5), (4,4), (5,3), (6,2).
   That is 5 outcomes.
   
   Step 3. Form the probability as favorable outcomes over total outcomes.
   P(sum = 8) = 5/36.
   
   Step 4. Check the result.
   Count sums systematically: for each first-die value a from 1 to 6, the second die must be 8 - a, which must lie between 1 and 6. For a = 1 the required value is 7 (not allowed), a = 2 gives 6, a = 3 gives 5, a = 4 gives 4, a = 5 gives 3, a = 6 gives 2. That gives 5 valid cases, matching Step 2.
   As a sanity check, 8 is the most likely sum for two dice only after 7, and 5/36 (about 0.139) is smaller than 6/36 for sum 7, which is the expected pattern. The probabilities for sums 2 through 12 are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 over 36, and the entry for 8 is 5, consistent with the result.
   
   The problem reads unambiguously as the sum of the two dice being exactly 8, with both dice fair and rolled once each.
   
   Final answer: 5/36

   459 tokens in, 506 out · 2.9 s · $0.0003 · 1 message on Pro · answered by anthropic/claude-haiku-5.5 via Anthropic · October 7, 2026

4. DeepSeek V4.1 Flash

   Passed: Final answer 5/36: right.

   1. Each die has 6 equally likely outcomes, so the total number of ordered outcomes for two dice is 6 * 6 = 36.
   
   2. The ordered pairs of dice rolls that add up to 8 are:
   (2,6), (3,5), (4,4), (5,3), (6,2).
   
   3. There are 5 such ordered pairs.
   
   4. Therefore, the probability is the number of favorable outcomes divided by the total number of outcomes:
   5/36.
   
   5. Check: The possible sums around 8 are 7 with 6 outcomes, 8 with 5 outcomes, and 9 with 4 outcomes, which fits the usual triangular pattern. Also, 5/36 is about 0.1389, a reasonable probability.
   
   Final answer: 5/36

   373 tokens in, 236 out (63 of them reasoning) · 0.9 s · $0.0003 · 1 message on Pro · answered by deepseek/deepseek-v4.1-flash via Together · October 8, 2026

5. GLM 5.3

   Passed: Final answer 5/36: right.

   Step 1: Find the total number of equally likely outcomes when rolling two dice.
   
   Each die has 6 faces, so total outcomes = 6 x 6 = 36.
   
   Step 2: List the outcomes where the sum is 8.
   
   (2,6), (3,5), (4,4), (5,3), (6,2) — that is 5 outcomes.
   
   Step 3: Compute the probability.
   
   P(sum = 8) = favorable / total = 5/36.
   
   Step 4: Check.
   
   Each pair listed indeed sums to 8 (e.g., 2+6=8, 4+4=8), and no others do (1 needs a 7, which is impossible; 7 needs a 1, impossible). The count 5 out of 36 is consistent with the known sum distribution, where 7 has 6 outcomes and 8 has 5.
   
   Final answer: 5/36

   355 tokens in, 202 out · 1.1 s · $0.0009 · 1 message on Pro · answered by z-ai/glm-5.3 via Friendli · September 28, 2026

6. GPT-6 Sol

   Passed: Final answer 5/36: right.

   1. Two dice have 6 × 6 = 36 equally likely outcomes.
   
   2. The outcomes that add up to 8 are (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2). There are 5.
   
   3. Probability = favorable outcomes / total outcomes.
   Probability = 5/36.
   
   4. Check: For the first die, only 2, 3, 4, 5, or 6 allows the second die to make a sum of 8. Each gives exactly one matching outcome, confirming the count of 5.
   
   Final answer: 5/36

   347 tokens in, 143 out · 1.9 s · $0.0011 · 1 message on Pro · answered by openai/gpt-6-sol via OpenAI · September 28, 2026

7. Gemini 3.8 Flash

   Passed: Final answer 5/36: right.

   Step 1: Find the total number of possible outcomes.
   Each fair six-sided die has 6 possible outcomes (1 through 6).
   Since the two dice are independent, the total number of outcomes is:
   6 * 6 = 36
   
   Step 2: List the outcomes where the sum of the two dice is 8.
   Let (Die 1, Die 2) represent an outcome:
   (2, 6) gives 2 + 6 = 8
   (3, 5) gives 3 + 5 = 8
   (4, 4) gives 4 + 4 = 8
   (5, 3) gives 5 + 3 = 8
   (6, 2) gives 6 + 2 = 8
   
   Step 3: Count the number of favorable outcomes.
   From the list above, there are 5 pairs that add up to 8.
   Note that rolling a 1 on either die cannot result in a sum of 8 because the maximum roll on the other die is 6 (1 + 6 = 7).
   
   Step 4: Calculate the probability.
   The probability is the number of favorable outcomes divided by the total number of outcomes:
   P(sum = 8) = 5 / 36
   
   Step 5: Check the result.
   The sums from rolling two dice range from 2 to 12.
   The number of ways to roll each sum is:
   Sum 2: 1 way
   Sum 3: 2 ways
   Sum 4: 3 ways
   Sum 5: 4 ways
   Sum 6: 5 ways
   Sum 7: 6 ways
   Sum 8: 5 ways
   Sum 9: 4 ways
   Sum 10: 3 ways
   Sum 11: 2 ways
   Sum 12: 1 way
   Total ways = 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36.
   The count for a sum of 8 is confirmed to be 5.
   The fraction 5/36 cannot be simplified further since 5 is prime and does not divide 36.
   
   Final answer: 5/36

   357 tokens in, 495 out · 2.8 s · $0.0011 · 1 message on Pro · answered by google/gemini-3.8-flash via Google AI Studio · September 28, 2026

8. GPT-6.1 Sol

   Passed: Final answer 5/36: right.

   1. Count the equally likely outcomes.
   Each die has 6 possible results, so the number of ordered pairs is:
   6 × 6 = 36
   
   2. List the outcomes whose numbers add up to 8.
   (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
   There are 5 favorable outcomes.
   
   3. Divide favorable outcomes by total outcomes.
   P(sum is 8) = 5/36
   
   4. Check the count.
   For each first-die result from 2 through 6, exactly one second-die result makes 8. A first-die result of 1 would require 7, which is impossible. This confirms 5 favorable outcomes out of 36.
   
   Final answer: 5/36

   347 tokens in, 171 out · 2.7 s · $0.0012 · 1 message on Pro · answered by openai/gpt-6.1-sol via OpenAI · September 29, 2026

9. DeepSeek V4 Pro

   Passed: Final answer 5/36: right.

   1. There are 6 possible outcomes for the first die and 6 for the second die.
   2. Total equally likely outcomes: 6 * 6 = 36.
   3. List the ordered pairs that add up to 8:
      (2,6), (3,5), (4,4), (5,3), (6,2).
   4. Count these favorable outcomes: 5.
   5. Probability = favorable outcomes / total outcomes = 5/36.
   6. Check: the number of ways to roll sums 2 through 12 are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. These add to 36, and the count for sum 8 is 5, so the result is consistent.
   
   Final answer: 5/36

   347 tokens in, 515 out (334 of them reasoning) · 3.4 s · $0.0021 · 1 message on Pro · answered by deepseek/deepseek-v4-pro-0813 via Wafer · October 8, 2026

10. Claude Haiku 4.5

    Passed: Final answer 5/36: right.

    I need to find the probability that two fair six-sided dice sum to 8.
    
    1. First, I'll find the total number of possible outcomes when rolling two dice.
       Each die has 6 faces, so there are 6 × 6 = 36 equally likely outcomes.
    
    2. Next, I'll count the favorable outcomes where the sum equals 8.
       I need to find all pairs (first die, second die) that add to 8:
       - (2, 6): 2 + 6 = 8 ✓
       - (3, 5): 3 + 5 = 8 ✓
       - (4, 4): 4 + 4 = 8 ✓
       - (5, 3): 5 + 3 = 8 ✓
       - (6, 2): 6 + 2 = 8 ✓
       
       That's 5 favorable outcomes.
    
    3. Now I'll calculate the probability.
       Probability = (Number of favorable outcomes) / (Total number of outcomes)
       Probability = 5/36
    
    4. Let me verify this makes sense by checking my list of pairs again:
       - For sum = 8, the pairs are: (2,6), (3,5), (4,4), (5,3), (6,2)
       - That's indeed 5 pairs out of 36 possible outcomes
       - Each pair has equal probability since both dice are fair
       - The fraction 5/36 cannot be simplified further since 5 and 36 share no common factors
    
    Final answer: 5/36

    365 tokens in, 390 out · 2.9 s · $0.0023 · 1 message on Pro · answered by anthropic/claude-haiku-4.5 via Anthropic · September 28, 2026

11. Claude Sonnet 5

    Passed: Final answer 5/36: right.

    1. Total outcomes when rolling two dice: 6 × 6 = 36, since each die has 6 faces independently.
    
    2. List outcomes that sum to 8:
    (2,6), (3,5), (4,4), (5,3), (6,2)
    
    3. Count favorable outcomes: 5
    
    4. Probability = favorable outcomes / total outcomes = 5/36
    
    5. Check: 5/36 ≈ 0.139, which is reasonable since 8 is near the most likely sum (7), so a probability slightly less than the max probability (6/36 for sum=7) makes sense.
    
    Final answer: 5/36

    457 tokens in, 248 out (47 of them reasoning) · 3.0 s · $0.0034 · 1 message on Pro · answered by anthropic/claude-sonnet-5 via Anthropic · September 28, 2026

12. Kimi K3

    Passed: Final answer 5/36: right.

    Step 1: Find the total number of possible outcomes.
    
    Each die has 6 faces, and the two dice are independent, so the total number of outcomes is:
    6 x 6 = 36
    
    Step 2: Find all outcomes where the two numbers add up to 8.
    
    List the ordered pairs (first die, second die):
    (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
    
    Step 3: Count the favorable outcomes.
    
    There are 5 such pairs.
    (Note: (2, 6) and (6, 2) are different outcomes because the dice are distinct; same for (3, 5) and (5, 3).)
    
    Step 4: Compute the probability.
    
    Probability = (favorable outcomes) / (total outcomes) = 5/36
    
    Step 5: Check the result.
    
    The probabilities for all possible sums (2 through 12) must add to 1:
    1/36 + 2/36 + 3/36 + 4/36 + 5/36 + 6/36 + 5/36 + 4/36 + 3/36 + 2/36 + 1/36 = 36/36 = 1
    The counts are consistent, and 5 favorable outcomes for a sum of 8 matches this pattern, so the answer checks out. As a decimal, 5/36 is about 0.139, or roughly 13.9%.
    
    Final answer: 5/36

    434 tokens in, 346 out (9 of them reasoning) · 29.8 s · $0.0035 · 1 message on Pro · answered by moonshotai/kimi-k3 via Wafer · September 28, 2026

13. Mistral Large 4

    Passed: Final answer 5/36: right.

    Step 1: Find the total number of possible outcomes when rolling two dice.
    The first die can show any of 6 numbers.
    The second die can show any of 6 numbers.
    Total outcomes = 6 × 6
    Total outcomes = 36
    This follows from the multiplication principle: each of the 6 results on the first die can be paired with each of the 6 results on the second die.
    
    Step 2: Find all outcomes where the two numbers add up to 8.
    We look for ordered pairs (first die, second die) whose sum is 8.
    (2, 6) since 2 + 6 = 8
    (3, 5) since 3 + 5 = 8
    (4, 4) since 4 + 4 = 8
    (5, 3) since 5 + 3 = 8
    (6, 2) since 6 + 2 = 8
    No other pairs work because if the first die shows 1, the second would need to show 7 (impossible on a six-sided die), and any value above 6 is impossible.
    Number of favorable outcomes = 5
    
    Step 3: Calculate the probability.
    Probability = (number of favorable outcomes) / (total number of outcomes)
    Probability = 5 / 36
    
    Step 4: Check the result.
    The total number of equally likely outcomes is 36, which is correct for two dice.
    The favorable outcomes are (2,6), (3,5), (4,4), (5,3), and (6,2), giving exactly 5 outcomes.
    As a decimal, 5/36 ≈ 0.1389 or about 13.9%.
    This is reasonable because the sum of 7 is the most likely outcome with probability 6/36, and 8 is adjacent to it with slightly lower probability 5/36.
    The fraction 5/36 is already in lowest terms.
    
    Final answer: 5/36

    351 tokens in, 2,568 out (1,796 of them reasoning) · 40.4 s · $0.0056 · 1 message on Pro · answered by mistralai/mistral-large-4-0 via Mistral · October 9, 2026

14. Claude Sonnet 5.5

    Passed: Final answer 5/36: right.

    Reading used: two distinguishable fair dice (so (2,6) and (6,2) count as different outcomes), each face equally likely.
    
    Step 1: Count the total number of outcomes.
    Each die has 6 faces, so the number of ordered pairs is
    6 × 6 = 36
    Each of these 36 outcomes is equally likely, so probability = (favorable outcomes)/36.
    
    Step 2: List the pairs (first die, second die) that sum to 8.
    If the first die is 1, the second would need to be 7, which is impossible.
    First die 2, second die 6: (2,6)
    First die 3, second die 5: (3,5)
    First die 4, second die 4: (4,4)
    First die 5, second die 3: (5,3)
    First die 6, second die 2: (6,2)
    
    Step 3: Count the favorable outcomes.
    There are 5 pairs.
    
    Step 4: Compute the probability.
    P(sum = 8) = 5/36
    
    Step 5: Check the result.
    The number of ways to get sums 2 through 12 is 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. These add up to 36, which matches the total. The sum 8 is the second sum after the peak at 7, so it has 5 ways, which agrees with our count.
    As a decimal, 5/36 ≈ 0.139, or about 13.9%, which is reasonable, since it is a bit less than the most likely sum (7, with probability 6/36 = 1/6).
    
    Final answer: 5/36

    459 tokens in, 499 out · 4.1 s · $0.0059 · 1 message on Pro · answered by anthropic/claude-sonnet-5.5 via Anthropic · September 28, 2026

15. GPT-6 Astra

    Passed: Final answer 5/36: right.

    1. Count all equally likely outcomes.
    Each die has 6 possible results, so the number of ordered pairs is:
    6 × 6 = 36
    
    2. Count the outcomes that add up to 8.
    The ordered pairs are:
    (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
    There are 5 favorable outcomes.
    
    3. Divide favorable outcomes by total outcomes.
    P(sum is 8) = 5/36
    
    4. Check the count.
    If the first die is 1, the second would need to be 7, which is impossible. For each first-die result from 2 through 6, exactly one second-die result gives a sum of 8. This confirms 5 favorable outcomes out of 36.
    
    Final answer: 5/36

    347 tokens in, 182 out · 2.9 s · $0.0063 · 1 message on Pro · answered by openai/gpt-6-astra via OpenAI · September 28, 2026

16. Gemini 3.1 Pro

    Passed: Final answer 5/36: right.

    1. Find the total number of possible outcomes when rolling two fair six-sided dice. 
    Since each die has 6 sides and they are rolled independently, you multiply the number of sides together:
    Total outcomes = 6 * 6
    Total outcomes = 36
    
    2. Find the number of successful outcomes where the numbers on the dice add up to exactly 8.
    We list all the possible pairs of numbers (Die 1, Die 2) that sum to 8, remembering that each die can only show numbers from 1 to 6:
    If Die 1 rolls a 2, Die 2 must roll a 6 (2 + 6 = 8).
    If Die 1 rolls a 3, Die 2 must roll a 5 (3 + 5 = 8).
    If Die 1 rolls a 4, Die 2 must roll a 4 (4 + 4 = 8).
    If Die 1 rolls a 5, Die 2 must roll a 3 (5 + 3 = 8).
    If Die 1 rolls a 6, Die 2 must roll a 2 (6 + 2 = 8).
    
    3. Count the successful outcomes.
    Looking at the list above, there are 5 pairs that add up to 8: (2,6), (3,5), (4,4), (5,3), and (6,2).
    Successful outcomes = 5
    
    4. Calculate the probability.
    The probability of an event is the number of successful outcomes divided by the total number of possible outcomes:
    Probability = 5 / 36
    
    5. Check the result. 
    We can verify the number of ways to get all possible sums from 2 to 12 to ensure we haven't missed any outcomes. The pattern of frequencies for sums from 2 to 12 goes: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. The sum of 8 is the seventh item in this list, which corresponds to 5. Adding all these frequencies gives 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36, which matches our total outcomes. The fraction 5/36 cannot be simplified further.
    
    Final answer: 5/36

    357 tokens in, 1,031 out (508 of them reasoning) · 8.9 s · $0.0065 · 1 message on Pro · answered by google/gemini-3.1-pro-preview via Google AI Studio · September 28, 2026

17. Grok 4.7

    Passed: Final answer 5/36: right.

    Step 1: Find the total number of equally likely outcomes.
    Each die has 6 faces and the rolls are independent, so
    6 * 6 = 36.
    
    Step 2: List the outcomes whose faces add to 8, keeping only faces from 1 to 6.
    (2, 6) because 2 + 6 = 8
    (3, 5) because 3 + 5 = 8
    (4, 4) because 4 + 4 = 8
    (5, 3) because 5 + 3 = 8
    (6, 2) because 6 + 2 = 8
    That is 5 outcomes. Faces outside 1 to 6, such as 7, are impossible.
    
    Step 3: Form the probability as favorable outcomes divided by total outcomes.
    5 / 36
    
    Step 4: Check by fixing the first die and requiring the second die to equal 8 minus the first, and to lie between 1 and 6.
    First die 2 gives second die 6, valid.
    First die 3 gives second die 5, valid.
    First die 4 gives second die 4, valid.
    First die 5 gives second die 3, valid.
    First die 6 gives second die 2, valid.
    First die 1 or anything above 6 is invalid.
    Again exactly 5 outcomes, so 5/36 is confirmed. It also matches the known count that a sum of 8 is one less common than a sum of 7.
    
    Final answer: 5/36

    1,586 tokens in, 1,431 out (1,101 of them reasoning) · 16.7 s · $0.0100 · 1 message on Pro · answered by x-ai/grok-4.7 via xAI · October 2, 2026

18. Claude Opus 5.5

    Passed: Final answer 5/36 (about 0.139): right.

    1. Count all possible outcomes.
    Each die has 6 faces, and the two dice are independent, so the number of equally likely ordered outcomes is
    6 × 6 = 36.
    I treat the dice as distinguishable (first die, second die), so (2, 6) and (6, 2) are different outcomes. This is what makes all 36 outcomes equally likely.
    
    2. List the outcomes that add up to 8.
    The first die must be at least 2, because the second die can show at most 6.
    (2, 6)
    (3, 5)
    (4, 4)
    (5, 3)
    (6, 2)
    That gives 5 favorable outcomes.
    
    3. Compute the probability.
    Probability = favorable outcomes / total outcomes
    = 5/36
    
    4. Check.
    The number of ways to roll each sum from 2 to 12 is 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.
    These add to 36, which matches the total from step 1.
    The sum 8 sits right after the peak at 7, so it has 5 ways, which agrees with the list in step 2.
    As a decimal, 5/36 ≈ 0.139, or about 13.9%.
    
    Final answer: 5/36 (about 0.139)

    459 tokens in, 464 out (59 of them reasoning) · 4.8 s · $0.0111 · 1 message on Pro · answered by anthropic/claude-opus-5.5 via Google · September 28, 2026

19. Claude Fable 5.1

    Passed: Final answer 5/36: right.

    Step 1: Count the total number of equally likely outcomes.
    Each die has 6 faces, and the two rolls are independent, so the number of ordered outcomes is
    6 * 6 = 36.
    
    Step 2: List the ordered pairs (first die, second die) whose sum is 8.
    (2, 6)
    (3, 5)
    (4, 4)
    (5, 3)
    (6, 2)
    Note that (1, 7) and (7, 1) are impossible since a die shows at most 6, so these 5 pairs are all of them.
    
    Step 3: Form the probability as favorable outcomes over total outcomes.
    P(sum = 8) = 5/36
    
    Check: The number of ways to roll a sum s on two dice is s - 1 for s from 2 to 7, and 13 - s for s from 7 to 12. For s = 8 that gives 13 - 8 = 5, matching the list above. Also, 5/36 is about 0.139, slightly less than the most likely sum's probability of 6/36 for a sum of 7, which makes sense since 8 is one step away from 7.
    
    Final answer: 5/36

    459 tokens in, 362 out · 5.8 s · $0.0227 · 1 message on Pro · answered by anthropic/claude-fable-5.1 via Google · September 28, 2026

• A linear equation
• A discount word problem
• A remainder of a large power
• A definite integral by parts
• AI math solver: try it free, and every model's results