Use the applet entitled Simulating the Probability of Rolling a 6 to explore the relationship between the proportion of sixes rolled on several rolls of a die and the theoretical probability of rolling a 6 on a fair die.
a. To simulate rolling a die one time, click on the Roll button on the screen while n = 1. The outcome of the roll appears in the list at the right, and the cumulative proportion of sixes for one roll is shown above the graph and as a point on the graph corresponding to 1. Click Reset and repeat the process with n = 1 several times. What are the possible values of the cumulative proportion of sixes for one roll of a die? Can the cumulative proportion of sixes for one roll of a die equal the theoretical probability of rolling a 6 on a fair die? Explain.
b. Set n = 10 and click the Roll button. Repeat this several times, resetting after each time. Record the cumulative proportion of sixes for each roll. Compare the cumulative proportions for n = 10 with those for n = 1 in part a. Which tend to be closer to the theoretical probability of rolling a 6 on a fair die?
c. Repeat part b for n = 1,000, comparing the cumulative proportions for n = 1,000 with those for n = 1 in part a and for n = 10 in part b.
d. On the basis of your results for parts a, b, and c, do you believe that we could justifiably conclude that a die is unfair because we rolled it 10 times and didn't roll any Sixes? Explain.