Explain how empirical probability can be used to determine whether a die is “loaded” (not a fair die).
Empirical probability is based on observations or data collected from experiments or real-world events. To determine whether a die is "loaded," or not fair, you can use the empirical probability by conducting a series of dice rolls and analyzing the resulting outcomes.
Here are the steps to use empirical probability to investigate a potentially loaded die:
1. Set up an experiment: Start by rolling the die a sufficient number of times to gather substantial data. The greater the number of rolls, the more reliable your conclusions will be.
2. Collect data: Record the outcomes of each roll, noting the face of the die that lands up. This data will form the basis of your analysis.
3. Calculate the relative frequency: Determine the relative frequency of each face by dividing the number of times it appears by the total number of rolls. For example, if you roll the die 100 times and the face '6' appears 20 times, then the relative frequency of '6' is (20/100) = 0.2 or 20%.
4. Compare the relative frequencies: Compare the relative frequencies of each face. In a fair die, each face should occur with equal probability, resulting in approximately equal relative frequencies. If you notice significant variations in the relative frequencies among the faces, it suggests that the die may be loaded.
5. Evaluate the results: Consider the extent and consistency of the variations in the relative frequencies. If the differences between the relative frequencies are substantial and consistently skewed towards specific faces, it indicates the die is likely loaded.
6. Repeat the experiment: To strengthen your conclusion, perform multiple iterations of the experiment, rolling the die again and collecting new data. If the results consistently show imbalanced relative frequencies, it provides further evidence of a loaded die.
Keep in mind that empirical probability provides evidence based on observed data, but it does not provide a definitive conclusion. It is always possible to have some variation due to chance, even with a fair die. For a more conclusive determination, you may consider additional statistical tests or consult experts in probability and dice analysis.