Question

Probabilities of throwing two dice. Make a computer program for throwing two dice a large number of times. Record the sum of the eyes each time and count how many times each of the possibilities for the sum \((2,3, \ldots, 12)\) appear. A dictionary with the sum as key and count as value is convenient here. Divide the counts by the total number of trials such that you get the frequency of each possible sum. Write out the frequencies and compare them with exact probabilities. (To find the exact probabilities, set up all the \(6 \times 6\) possible outcomes of throwing two dice, and then count how many of them that has a sum \(s\) for \(s=2,3, \ldots, 12\).) Name of program file: freq_2dice.py.

Short answer

Simulate dice, tally sums, normalize results, calculate exact probabilities, and compare.

Step-by-step solution

Setting up the Dice Simulation

Start by writing a Python program to simulate the throwing of two dice a large number of times. Use the random module to generate random numbers between 1 and 6 for each die. Store the results of the dice throws as the sum of the two numbers.

Recording the Sum Frequencies

Create a dictionary to keep track of the frequency of each possible sum from 2 to 12. Initialize keys for each possible sum with a count of zero. For each pair of dice throws, calculate the sum and increment the corresponding count in the dictionary.

Calculating Relative Frequencies

After running the simulation for all trials, calculate the frequency of each sum. Divide the count for each sum by the total number of trials to get the relative frequencies. This normalization step will convert absolute counts into probabilities.

Computing Exact Probabilities

Manually compute the exact probabilities by considering all possible outcomes (36 total combinations) when two dice are thrown. Count how many outcomes produce each sum from 2 to 12. Divide the count for each sum by 36 to obtain its exact probability.

Comparing Results

Compare the relative frequencies obtained from the simulation with the exact probabilities calculated manually. Observe any discrepancies which could be due to the randomness of the simulation and the number of trials.

Key concepts

Python Programming

Python is a versatile language that can be used for simulating dice rolls and analyzing their outcomes. This exercise involves using Python to simulate a situation where you throw two dice many times. You start by importing the `random` module to generate random numbers between 1 and 6.

Each "throw" involves generating two numbers (one for each die) and adding them together to get the total number of "eyes" showing. These commands are simple yet powerful, allowing you to completely simulate the behavior of dice. Once the dice simulation is rolling, Python can also help in other crucial steps like storing results in a dictionary and analyzing the frequency of sums. Using loops and conditionals in Python, you can easily repeat experiments, gather results, and then dive into analysis.

Dice Probability

Dice probability revolves around the various outcomes that can occur when throwing dice. In our exercise, we are particularly interested in the sum of the two dice, which can range from 2 to 12. Each of these sums can occur in a certain number of ways, and understanding this is integral to computing probabilities.

For example:

• The sum of 2 can only be achieved with a pair of ones, making it less likely.
• The sum of 7 can happen in several ways: (1+6, 2+5, 3+4, etc.), making it more probable.

By listing all combinations, totaling 36, you can determine the theoretical probability of each sum by dividing the number of ways the sum can occur by 36. This probability gives you an exact measure of how likely each sum is before running the program simulation.

Random Number Generation

Random number generation is the core of this simulation exercise. It involves producing sequences of numbers that mimic the randomness of dice. In Python, this is achieved using the `random.randint()` function, generating a random integer between a specified range, like 1 and 6 in this case.

Each call to `random.randint(1, 6)` is like physically rolling a die, with each number representing a potential outcome on the die faces. This randomness is fundamental in accurately simulating how dice behave in real life. By using this built-in randomness, Python can help demonstrate the laws of probability with simulations that closely reflect actual outcomes of repeated random events.

Data Analysis in STEM Education

Data analysis is an essential skill in STEM education, and exercises like this provide a practical application. Here, we analyze data generated from simulated dice rolls to understand probability better. After collecting data from numerous dice rolls, you store these results in a dictionary. This dictionary offers a simple structure for organizing and retrieving the frequency of each sum.

Once all data are collected, you'll compute relative frequencies by dividing each sum's count by the total number of rolls. This process transforms raw data into more meaningful insights, showing how often each result occurs compared to others. By comparing these simulated frequencies with theoretical probabilities, students improve their understanding of probability theory and its applications in real-world situations.