\n ## Text Document: Prompt & Exemplars for Mathematical Problem Solving ### Overview The image presents a text document outlining a prompt for a Large Language Model (LLM) focused on solving mathematical problems. It includes instructions, a sample problem, and several "Self-generated Exemplars" demonstrating the desired input/output format. The document appears to be designed to guide the LLM in recalling relevant problems and then solving a new problem in a specific style. ### Components/Axes The document is structured into sections: * **Prompt:** The initial instruction given to the LLM. * **# Problem:** The initial mathematical problem to be solved. * **# Instructions:** Guidelines for the LLM's response. * **# Relevant Problems:** Instructions for generating related problems. * **# Solve the Problem:** Instructions for solving the initial problem. * **Output by GPT3.5-turbo:** A section showcasing example outputs. * **Self-generated Exemplars:** A series of example problems and solutions. ### Detailed Analysis or Content Details **Prompt:** "Your task is to tackle mathematical problems. When presented with a math problem, recall relevant problems as examples. Afterward, proceed to solve the initial problem." **# Problem:** "An airline serves a dinner to all the passengers on an airplane. They get their choice of steak or fish. Three steak meals and three fish meals are set aside for the six-member crew. If the meals are distributed to the crew members randomly, what is the probability that both pilots get the fish?" **# Instructions:** * "Recall three examples of math problems that are relevant to the initial problem. Your problems should be distinct from each other and from the initial problem (e.g., involving different numbers and names). For each problem: * After “Q: ", describe the problem * After “A: ", explain the solution and enclose the ultimate answer in \boxed{}" * "# Solve the Problem:" * "Q: Copy and paste the initial problem here." * "A: Explain the solution and enclose the ultimate answer in \boxed{}" **Output by GPT3.5-turbo:** **Relevant Problems:** * **Q:** In a bag, there are 5 red balls and 3 blue balls. If two balls are drawn at random without replacement, what is the probability that both balls are red? **A:** The total number of balls is 8. The probability of drawing a red ball on the first draw is 5/8. After one red ball is drawn, there are 4 red balls left out of 7 total balls. Therefore, the probability of drawing a second red ball is 4/7. The probability of both events happening is the product of the individual probabilities, so the probability of drawing two red balls is (5/8) * (4/7) = \boxed{frac{5}{14}}. * **Q:** A box contains 10 red marbles and 5 blue marbles. If three marbles are drawn at random without replacement, what is the probability that all three marbles are red? **A:** The total number of marbles is 15. The probability of drawing a red marble on the first draw is 10/15. After one red marble is drawn, there are 9 red marbles left out of 14 total marbles. Therefore, the probability of drawing a second red marble is 9/14. After two red marbles are drawn, there are 8 red marbles left out of 13 total marbles. Therefore, the probability of drawing a third red marble is 8/13. The probability of all three events happening is the product of the individual probabilities, so the probability of drawing three red marbles is (10/15) * (9/14) * (8/13) = \boxed{frac{24}{91}}. * **Q:** A restaurant has 7 cappuccinos and 4 lattes. If two drinks are drawn at random without replacement, what is the probability that both drinks are cappuccinos? **A:** The total number of drinks is 11. The probability of drawing a cappuccino on the first draw is 7/11. After one cappuccino is drawn, there are 6 cappuccinos left out of 10 total drinks. Therefore, the probability of drawing a second cappuccino is 6/10. The probability of both events happening is the product of the individual probabilities, so the probability of drawing two cappuccinos is (7/11) * (6/10) = \boxed{frac{21}{55}}. **# Solve the Problem:** * **Q:** An airline serves a dinner to all the passengers on an airplane. They get their choice of steak or fish. Three steak meals and three fish meals are set aside for the six-member crew. If the meals are distributed to the crew members randomly, what is the probability that both pilots get the fish? * **A:** There are a total of 6 meals, 3 steak and 3 fish. The total number of ways to distribute the 6 meals to the 6 crew members is 6!. However, since the steak meals are indistinguishable and the fish meals are indistinguishable, we need to divide by 3! * 3!. So the total number of ways to distribute the meals is 6! / (3! * 3!) = 20. Now, let's consider the case where both pilots get fish. This means there is one fish meal left to distribute among the remaining 4 crew members, along with 3 steak meals. The number of ways to distribute these remaining meals is 5! / (1! * 3!) = 20. The number of ways to choose 2 fish meals for the pilots is \binom{3}{2} = 3. The remaining 1 fish meal and 3 steak meals can be distributed among the remaining 4 crew members in \binom{5}{1} = 5 ways. So, the number of favorable outcomes is 3 * 5 = 15. The probability that both pilots get fish is the number of favorable outcomes divided by the total number of possible outcomes. The total number of ways to assign the meals is \binom{6}{3} = 20. The number of ways to assign the meals such that both pilots get fish is \binom{4}{1} = 4. Therefore, the probability is 4/20 = 1/5. Alternatively, the probability that the first pilot gets fish is 3/6 = 1/2. Given that the first pilot gets fish, the probability that the second pilot gets fish is 2/5. Therefore, the probability that both pilots get fish is (1/2) * (2/5) = \boxed{1/5}. ### Key Observations The document demonstrates a clear pattern of problem-solving: providing relevant examples followed by a solution to the target problem. The examples focus on probability calculations involving combinations and permutations. The solutions are presented in a step-by-step manner, with the final answer enclosed in a \boxed{} box. ### Interpretation This document is a training example for an LLM, specifically designed to improve its ability to solve mathematical problems by analogy. The "Relevant Problems" section aims to prime the model with similar problem structures, enabling it to apply the same reasoning to the target problem. The consistent formatting (Q: / A: structure, \boxed{} for answers) reinforces a specific output style. The examples all involve probability, suggesting the LLM is being trained to specialize in this area of mathematics. The document highlights the importance of breaking down complex problems into smaller, manageable steps and clearly explaining the reasoning behind each step. The use of LaTeX-style formatting (\frac{}) suggests the model is expected to handle mathematical notation.