4. Probability
Basic Concepts of Probability
3:23 minutesProblem 3.1.97
Textbook Question
97. Rolling a Pair of Dice You roll a pair of six-sided dice and record the sum.
a. List all of the possible sums and determine the probability of rolling each sum.
b. Use technology to simulate rolling a pair of dice and record the sum 100 times. Make a tally of the 100 sums and use these results to list the probability of rolling
each sum.
c. Compare the probabilities in part (a) with the probabilities in part (b). Explain any similarities or differences.
Verified step by step guidance1
Step 1: Understand the problem. You are rolling two six-sided dice and recording the sum. Each die has faces numbered 1 through 6, so the possible sums range from 2 (1+1) to 12 (6+6). Part (a) asks for the theoretical probabilities, part (b) involves simulating the rolls, and part (c) requires comparing the theoretical and simulated probabilities.
Step 2: For part (a), list all possible sums and calculate their probabilities. To do this, consider all possible combinations of the two dice. For example, the sum of 2 can only occur if both dice show 1, while the sum of 7 can occur in multiple ways (e.g., 1+6, 2+5, 3+4, etc.). Count the number of ways each sum can occur and divide by the total number of outcomes (6 × 6 = 36). Use the formula: , where f(x) is the frequency of the sum x.
Step 3: For part (b), use technology (e.g., a spreadsheet, Python, or a statistical software) to simulate rolling two dice 100 times. Generate random numbers between 1 and 6 for each die, add them together to get the sum, and record the results. Create a tally of how many times each sum occurs. Then calculate the experimental probability for each sum using the formula: , where f(x) is the frequency of the sum x in the simulation.
Step 4: For part (c), compare the theoretical probabilities from part (a) with the experimental probabilities from part (b). Look for similarities and differences. Discuss factors such as sample size (100 rolls) and randomness, which can cause the experimental probabilities to deviate slightly from the theoretical probabilities.
Step 5: Summarize your findings. Explain how increasing the number of simulations (e.g., rolling the dice more than 100 times) would likely make the experimental probabilities closer to the theoretical probabilities due to the Law of Large Numbers. Highlight any patterns or trends observed in the comparison.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability Distribution
A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In the context of rolling two dice, the sums can range from 2 to 12, and each sum has a specific probability based on the number of combinations that can produce it. Understanding this distribution is essential for calculating and comparing the theoretical probabilities with experimental results.
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Simulation
Simulation is a technique used to model the behavior of a system by generating random samples. In this case, simulating the rolling of two dice 100 times allows for the collection of empirical data on the sums rolled. This method helps to visualize and analyze the outcomes, providing a practical approach to understanding probability through real-world experimentation.
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Empirical vs. Theoretical Probability
Empirical probability is based on observed data from experiments, while theoretical probability is calculated based on the expected outcomes of a random process. In this question, part (a) involves calculating the theoretical probabilities of rolling each sum, while part (b) focuses on the empirical probabilities obtained from the simulation. Comparing these two types of probabilities can reveal insights into the accuracy of theoretical models and the variability inherent in random processes.
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