Question

(Dice Rolling) Write an application to simulate the rolling of two dice. The application should use an object of class Random once to roll the first die and again to roll the second die. The sum of the two values should then be calculated. Each dic can show an integer value from 1 to \(6,\) so the sum of the values will vary from 2 to \(12,\) with 7 being the most frequent \(\operatorname{sum},\) and 2 and 12 the least frequent. Figure 7.30 shows the 36 possible combinations of the two dice. Your application should roll the dice 36,000 times. Use a one-dimensional array to tally the number of times each possible sum appears. Display the results in tabular format. Determine whether the totals are reasonable (c.g., there are six ways to roll a \(7,\) so approximately one-sixth of the rolls should be 7 ).

Short answer

Simulate rolling two dice, tally sum occurrences, and verify results against expected frequencies.

Step-by-step solution

Initialize Random and Array

Create an instance of the Random class which will be used to simulate the dice rolls. Also, initialize a one-dimensional array of size 11 to zero, which will track the count of each sum from 2 to 12.

Simulate Dice Rolls

Set up a loop that runs 36,000 times to simulate rolling two dice. In each iteration, use Random to generate a value for each die by adding 1 to Random.nextInt(6), which gives a number between 1 and 6.

Calculate Dice Sum

For each pair of dice values obtained in the loop, compute their sum. This will be a number between 2 and 12. Increment the corresponding index in your array by 1 to tally the occurrence of each sum.

Display Results in a Table

After completing the 36,000 rolls, iterate over your array and print out the sum of the dice rolls alongside the count of how many times each sum appeared. This will allow you to see which sums occur the most frequently.

Analyze Results

Check whether the distribution of dice sums follows the expected pattern: approximately one-sixth of the rolls should result in the sum being 7, while sums of 2 and 12 should occur less frequently. This checks if the outcome is statistically reasonable.

Key concepts

Random Class Usage

The Random class in Java is essential for simulating random events, such as dice rolls. To generate random numbers that represent the outcome of dice, you use the Random class to produce values. Each call to Random.nextInt(6) generates a random integer between 0 and 5. By adding 1, you shift the range to 1 through 6, accurately simulating a die roll. This approach is straightforward and ensures each possible dice value is equally probable, mimicking a real die's behavior.

Using the Random class not only simplifies code but also ensures efficiency by reusing the same Random instance throughout the simulation. Creating just one Random object improves performance and maintains consistency, essential for a large number of simulations like rolling dice 36,000 times.

Array Tallying

After rolling the dice, tallying each sum is crucial for analyzing the results. In this simulation, an array of size 11 is used. This array keeps track of how often each sum, between 2 and 12, occurs. The reason for size 11 is because you are monitoring results for sums 2 to 12, creating 11 possible outcomes. Index 0 in the array corresponds to sum 2, index 1 to sum 3, and so on up to index 10 for sum 12.

Whenever a pair of dice is rolled and their sum calculated, you simply increase the corresponding index in this array by 1. This counts each sum's appearance during the simulation. Using arrays for tallying is efficient as it allows constant-time complexity, meaning each increment operation completes in the same time, regardless of the number of times it is performed.

• Efficient memory usage to store simple integers.
• Direct index access makes tallying fast and reliable.

Statistical Analysis

Statistical analysis is at the core of understanding the results of this dice simulation. By analyzing how frequently each sum appears, you can judge the accuracy and fairness of the dice roll simulation. This is achieved by comparing the expected frequencies of each sum to the observed frequencies.

For example, with two dice, there are six different combinations to obtain a sum of 7 (like 1+6, 2+5, etc.), meaning it should appear approximately one-sixth of the time. Similarly, sums like 2 (1+1) or 12 (6+6) only have one combination, so they should appear far less frequently. Calculating the percentage of times each sum appears lets you compare these with theoretical probabilities, thus diagnosing potential biases or errors in the simulation.

• Utilize percentages to compare expected versus actual outcomes.
• Check for abnormalities to improve simulation accuracy.

Probability Distribution

The probability distribution for the sums of two dice is a key concept in this simulation exercise. When you roll two dice, each possible outcome varies in probability, forming a distribution of sums that is not uniform. Understanding this distribution helps in predicting and verifying the results of your roll simulation.

The sum 7, being the midpoint, has the highest probability because it results from the most combinations (e.g., 1+6, 2+5, etc.). This makes 7 the mode of the distribution. The edge sums, like 2 and 12, are much less probable as they arise from only one combination each.

When designing or analyzing rolling dice simulations, ensuring your observed results match this theoretical probability distribution provides validation. This confirms the simulation is running correctly and reflects real-world randomness accurately. By confirming that outcomes align with the expected probabilities, you gain confidence in the simulation’s effectiveness.