| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | C. \frac{5}{54} |
C |
| deepseekr1 | To determine the probability that three dice show prime numbers and three show composite numbers when rolling six 6-sided dice: 1. **Identify primes and composites on a die**: - Primes: 2, 3, 5 (3 numbers). - Composites: 4, 6 (2 numbers). - 1 is neither and must be excluded for the condition to hold. 2. **Calculate favorable outcomes**: - Choose 3 dice out of 6 to show primes: \( \binom{6}{3} = 20 \). - Each prime die has 3 possibilities, and each composite die has 2 possibilities: \[ \text{Favorable outcomes} = 20 \times 3^3 \times 2^3 = 20 \times 27 \times 8 = 4320. \] 3. **Total possible outcomes**: - Rolling six dice: \( 6^6 = 46656 \). 4. **Compute the probability**: \[ \text{Probability} = \frac{4320}{46656} = \frac{5}{54}. \] **Answer**: C. \frac{5}{54} |
C |
| gpto1 | C | C |
| GPT-4o | To determine the probability that three out of six 6-sided dice show prime numbers and the remaining three show composite numbers, we can follow these steps: 1. **Identify the prime and composite numbers on a 6-sided die:** - Prime numbers: 2, 3, and 5 - Composite numbers: 4 and 6 (Note: 1 is neither prime nor composite) 2. **Calculate the probability of rolling a prime number:** - There are 3 prime numbers (2, 3, 5) out of 6 possible outcomes. - Probability of rolling a prime number: \( \frac{3}{6} = \frac{1}{2} \) 3. **Calculate the probability of rolling a composite number:** - There are 2 composite numbers (4, 6) out of 6 possible outcomes. - Probability of rolling a composite number: \( \frac{2}{6} = \frac{1}{3} \) 4. **Calculate the probability of getting exactly 3 prime numbers and 3 composite numbers:** - This is a binomial probability problem where we need to find the probability of getting exactly 3 primes (success) in 6 rolls. - The binomial coefficient \( \binom{6}{3} \) represents the number of ways to choose 3 successes (prime numbers) out of 6 trials (dice rolls): \[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = 20 \] - The probability of getting 3 primes and 3 composites: \[ P(X = 3) = \binom{6}{3} \left( \frac{1}{2} \right)^3 \left( \frac{1}{3} \right)^3 \] \[ P(X = 3) = 20 \cdot \left( \frac{1}{2} \right)^3 \cdot \left( \frac{1}{3} \right)^3 \] \[ P(X = 3) = 20 \cdot \frac{1}{8} \cdot \frac{1}{27} \] \[ P(X = 3) = 20 \cdot \frac{1}{216} \] \[ P(X = 3) = \frac{20}{216} = \frac{5}{54} \] Thus, the probability that three of the dice show prime numbers and the rest show composite numbers is \( \frac{5}{54} \). The correct answer is: C. \(\frac{5}{54}\) |
C |
| llama3.1-70B-Instruct | Answer: C |
C |
| llama3.1-405B-Instruct | C. \frac{5}{54} |
C |
| DeepSeek-Coder-V2 | C. \frac{5}{54} | C |
| Qwen2-72B-Instruct | A. \frac{125}{2916} | C |
| baichuan4 | A | C |
| doubao | Prime numbers on a 6-sided die are 2, 3, 5. The probability of getting a prime number is 3/6 = 1/2. Composite numbers on a 6-sided die are 4, 6. The probability of getting a composite number is 2/6 = 1/3. The number of ways to choose 3 dice out of 6 to show prime numbers is C(6,3) = 20. The probability of getting 3 prime numbers is (1/2)^3 = 1/8. The probability of getting 3 composite numbers is (1/3)^3 = 1/27. So the overall probability is 20 * (1/8) * (1/27) = 5/54. So the answer is C. |
C |
| minimax | To solve this problem, we need to calculate the probability of rolling exactly three prime numbers and three composite numbers with six 6-sided dice. First, let's identify the prime and composite numbers on a 6-sided die: - Prime numbers on a die: 2, 3, 5 - Composite numbers on a die: 4, 6 (1 is neither prime nor composite, and 6 is composite because it has divisors other than 1 and itself) There are 3 prime numbers and 2 composite numbers on a die. Now, we need to calculate the number of ways to choose 3 dice out of 6 to show prime numbers. This can be done using combinations: Number of ways to choose 3 dice for prime numbers = 6 choose 3 = C(6,3) = 6! / (3! * (6-3)!) = 20 For each of these combinations, the three dice showing prime numbers can each land on any of the 3 prime numbers, and the three dice showing composite numbers can each land on any of the 2 composite numbers. So, the total number of favorable outcomes is: 20 (ways to choose which dice show prime numbers) * 3^3 (ways to roll a prime number on each of the three dice) * 2^3 (ways to roll a composite number on each of the remaining three dice) = 20 * 27 * 8 = 4320 The total number of possible outcomes when rolling six 6-sided dice is 6^6, since each die has 6 faces and there are 6 dice. So, the probability P of rolling exactly three prime numbers and three composite numbers is: P = Number of favorable outcomes / Total number of possible outcomes P = 4320 / 6^6 P = 4320 / 46656 P = 5 / 54 The correct answer is C. \frac{5}{54}. |
C |
| kimi | To solve this problem, we need to calculate the probability of rolling three dice that show prime numbers (2, 3, or 5) and three dice that show composite numbers (4, 6). Since a die has six sides, the prime numbers are 2, 3, and 5, and the composite numbers are 4 and 6. First, let's calculate the probability of rolling a prime number on a single die: - There are 3 prime numbers out of 6 possible outcomes, so the probability of rolling a prime number is \( \frac{3}{6} = \frac{1}{2} \). Next, let's calculate the probability of rolling a composite number on a single die: - There are 2 composite numbers out of 6 possible outcomes, so the probability of rolling a composite number is \( \frac{2}{6} = \frac{1}{3} \). Now, we need to find the probability of rolling exactly three prime numbers and three composite numbers out of six dice. This is a combination problem where we choose 3 dice to show prime numbers out of 6 dice. The number of ways to choose 3 dice out of 6 is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. So, the number of ways to choose 3 dice to show prime numbers is \( C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \). The probability of rolling three prime numbers and three composite numbers is the product of the probability of rolling a prime number three times and the probability of rolling a composite number three times, multiplied by the number of ways to choose which dice show prime numbers. So, the probability is: \[ P = C(6, 3) \times \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{3}\right)^3 \] \[ P = 20 \times \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{3}\right)^3 \] \[ P = 20 \times \frac{1}{8} \times \frac{1}{27} \] \[ P = 20 \times \frac{1}{216} \] \[ P = \frac{20}{216} \] \[ P = \frac{5}{54} \] Therefore, the correct answer is: C. \( \frac{5}{54} \) |
C |
| wenxinyiyan | C | C |
| zhipu | To solve this problem, we need to calculate the probability that exactly three out of six 6-sided dice show prime numbers and the remaining three show composite numbers. First, let's identify the prime and composite numbers on a 6-sided die: - Prime numbers: 2, 3, 5 (3 primes) - Composite numbers: 4, 6 (2 composites) - Note: 1 is neither prime nor composite. The probability of rolling a prime number on a single die is: \[ P(\text{prime}) = \frac{3}{6} = \frac{1}{2} \] The probability of rolling a composite number on a single die is: \[ P(\text{composite}) = \frac{2}{6} = \frac{1}{3} \] We need to find the probability that exactly three dice show prime numbers and three dice show composite numbers. The number of ways to choose 3 dice out of 6 to show prime numbers is given by the combination: \[ \binom{6}{3} = 20 \] For each of these ways, the probability that the chosen 3 dice show prime numbers and the other 3 dice show composite numbers is: \[ \left( \frac{1}{2} \right)^3 \times \left( \frac{1}{3} \right)^3 = \frac{1}{8} \times \frac{1}{27} = \frac{1}{216} \] Therefore, the total probability is: \[ 20 \times \frac{1}{216} = \frac{20}{216} = \frac{5}{54} \] So, the correct answer is: C. \(\frac{5}{54}\) |
C |
| gpto1 | C | C |
| deepseekv3 | To solve this problem, let's break it down step by step. ### Step 1: Identify Prime and Composite Numbers on a 6-sided Die A standard 6-sided die has the numbers {1, 2, 3, 4, 5, 6}. - **Prime numbers**: 2, 3, 5 - **Composite numbers**: 4, 6 - **Neither prime nor composite**: 1 ### Step 2: Probabilities of Prime and Composite Numbers - Probability of rolling a prime number: \( P(\text{Prime}) = \frac{3}{6} = \frac{1}{2} \) - Probability of rolling a composite number: \( P(\text{Composite}) = \frac{2}{6} = \frac{1}{3} \) ### Step 3: Number of Ways to Choose Which Dice Show Prime Numbers When rolling 6 dice, the number of ways to choose 3 dice to show prime numbers (and the remaining 3 to show composite numbers) is given by the combination formula: \[ \binom{6}{3} = 20 \] ### Step 4: Calculate the Probability The probability for one specific arrangement (e.g., first 3 dice show prime numbers and the last 3 show composite numbers) is: \[ \left( \frac{1}{2} \right)^3 \cdot \left( \frac{1}{3} \right)^3 = \frac{1}{8} \cdot \frac{1}{27} = \frac{1}{216} \] Since there are 20 such arrangements, the total probability is: \[ 20 \cdot \frac{1}{216} = \frac{20}{216} = \frac{5}{54} \] ### Step 5: Select the Correct Answer The correct answer is: **C. \(\frac{5}{54}\)** |
C |