For a single dice roll, there is a 1/6 chance that any particular number will appear. For a pair of dice, each specific combination of numbers has a probability of 1/36 occurring. Most total values of two dice can occur more than one way. As a test of random probability theory, a student decides to roll a pair of six-sided dice 300 times and tabulate the results. She tabulates the number of times each different total value of the two dice occurs. Her results are the following:

Total Value of Two Dice      Number of Times Rolled
               2                                          7
               3                                         11
               4                                         23
               5                                         36
               6                                         42
               7                                         53
               8                                         40
               9                                         38
              10                                        30
              11                                        12
              12                                         8
           TOTAL                                   300

The student tells you that her results fail to prove that random chance is the explanation for the outcome of this experiment. Is she correct or incorrect? Support your answer.

Verified step by step guidance

1. Calculate the expected probability for each total value of two dice. For example, the probability of rolling a total of 7 is the highest because there are more combinations (6) that result in 7.
2. Determine the expected frequency for each total value by multiplying the probability of each total by the total number of rolls (300).
3. Compare the expected frequencies with the observed frequencies from the student's results.
4. Use a statistical test, such as the chi-square test, to determine if the differences between observed and expected frequencies are statistically significant.
5. Interpret the results of the statistical test to decide if the observed frequencies could be due to random chance or if there is a significant deviation.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of events occurring. In the context of rolling dice, it quantifies the chances of obtaining specific outcomes. For example, when rolling two six-sided dice, the total number of possible outcomes is 36, and each combination has a defined probability. Understanding these probabilities is essential for analyzing the student's results.

Randomness and Variability

Randomness refers to the unpredictability of outcomes in an experiment, while variability indicates the differences in results that can occur due to chance. In rolling dice, even with a fair setup, the results can vary significantly over a limited number of trials. This concept is crucial for interpreting the student's findings, as random chance can lead to unexpected distributions of outcomes, which may not align with theoretical probabilities.

Statistical Significance

Statistical significance is a measure that helps determine whether the results of an experiment are likely due to chance or if they reflect a true effect. In the context of the student's dice rolls, analyzing the frequency of each total value against expected probabilities can reveal whether the observed results deviate significantly from what would be expected by random chance. This analysis is key to supporting or refuting the student's claim about randomness.