Question

Write a program that simulates the rolling of two dice. The program should use rand to roll the first die and should use rand again to roll the second die. The sum of the two values should then be calculated. [ Note: Each die can show an integer value from 1 to \(6,\) so the sum of the two values will vary from 2 to \(12,\) with 7 being the most frequent sum and 2 and 12 being the least frequent sums.] Figure 7.32 shows the 36 possible combinations of the two dice. Your program should roll the two dice 36,000 times. Use a onedimensional array to tally the numbers of times each possible sum appears. Print the results in a tabular format. Also, determine if the totals are reasonable (i.e., there are six ways to roll a \(7,\) so approximately one-sixth of all the rolls should be 7 ).

Short answer

The program rolls two dice 36,000 times, tallies and prints the sums, showing that the sum 7 appears about 6,000 times, as expected.

Step-by-step solution

Initialize the Program

First, we will initialize the program by importing the necessary library. In our case, we will use the `random` library to generate random numbers, which simulate the rolling of the dice. We also set up an array `dice_sums` of size 13 (from 0 to 12, but we won't use index 0 and 1) initialized with zeros to keep a count of each possible sum of the dice.

Roll the Dice

We will use a loop to simulate the rolling of two dice a total of 36,000 times. Inside the loop, use `random.randint(1, 6)` twice to simulate the rolling of two dice. Add the results of these rolls to get the sum of the two dice.

Tally the Results

For each iteration, after calculating the sum of the two dice, increase the tally of the corresponding index in the `dice_sums` array. For example, if the sum is 7, increase `dice_sums[7]` by one.

Print Results in Tabular Format

After the loop has completed 36,000 iterations, print out the results stored in `dice_sums`. Use a formatted print statement to iterate over the indices 2 through 12 (as only these sums are possible with two dice) and print each sum along with its frequency count.

Analyze the Results

Analyze the tallied results to ensure they align with expected probabilities. For instance, since there are 6 ways to roll a sum of 7, check if the number of times 7 appeared is approximately one-sixth of 36,000. This should be around 6,000 times.

Key concepts

Random Number Generation

Creating a program that simulates dice rolling involves generating random numbers, as each face of a die corresponds to a random integer between 1 and 6. In this exercise, the `random` library in Python plays a crucial role. It provides a function called `random.randint(a, b)`, which returns a random integer N such that \(a \leq N \leq b\).

In the context of rolling dice, this function is used twice within a loop to randomly determine the values of two dice. Each call to `random.randint(1, 6)` mimics the throw of a single die, producing a random outcome from the set of possible die faces. By calling it twice, you simulate rolling two dice, which are then summed to get a result ranging from 2 to 12. Consistently generating random numbers thus allows us to realistically mimic the chance element inherent to dice games.

One-Dimensional Arrays

One-dimensional arrays are simple data structures that store a list of items, with each item accessible via an index. In our dice simulation program, a one-dimensional array, `dice_sums`, is used to keep track of how often each possible sum of two dice appears.

The array is initialized with 13 elements, since the index represents possible sums from 0 to 12. However, indexes 0 and 1 are not used because such sums are not possible with two six-sided dice. The remaining indices, 2 through 12, are utilized to count occurrences of each sum. This helps in efficiently organizing and updating scores each time the dice are rolled, allowing us to easily tally the frequency of each result after the simulation completes.

Frequency Tally

In data analysis, a frequency tally is a technique for counting how often specific data points occur within a set. In this dice simulation, the frequency tally is accomplished by incrementing the indices of the `dice_sums` array.

Each roll of the dice may produce a sum between 2 to 12. As each roll result is calculated, the program increases the corresponding index of the `dice_sums` array by one. This marks one more occurrence of that particular sum. By iterating over 36,000 rolls, the program builds a distribution of frequencies which reflects the probability distribution of rolling two dice, recording how many times each sum appears over all trials.

Ultimately, this frequency count not only stores useful data but also sets the stage for further statistical analysis.

Probability Analysis

Probability analysis involves interpreting the frequency of outcomes to infer likelihoods. With the dice simulation results, analyzing the accumulated data helps verify the program's accuracy against expected probabilities.

For example, rolling a 7 is the most common outcome since it can be formed by six different dice pairings \((1+6, 2+5, 3+4, 4+3, 5+2, 6+1)\). As such, one would expect the frequency of rolling a 7 in 36,000 dice throws to approximate one-sixth of total rolls, or about 6,000. Similarly, sums of 2 and 12, only achievable with (1+1) and (6+6) respectively, should appear far less frequently.

The validity of simulation outcomes can be confirmed by comparing collected data to these theoretical expectations. If the outcomes match reasonably well with the expected probabilities, the program is likely simulating the dice rolls accurately, thus completing an insightful probability analysis.