The law of large numbers. Suppose that you roll two balanced dice and look atthe spots

Chapter 0, Problem 11.29

(choose chapter or problem)

The law of large numbers. Suppose that you roll two balanced dice and look atthe spots on the up-faces. There are 36 possible outcomes, displayed in Figure10.2 (page 251). Because the dice are balanced, all 36 outcomes are equally likely.The average number of spots is 7. This is the population mean for the idealizedpopulation that contains the results of rolling two dice forever. The law of largenumbers says that the average x from a finite number of rolls gets closer and closerto 7 as we do more and more rolls.(a) Click More dice once in the Law of Large Numbers applet to get two dice.APPLETClick Show mean to see the mean 7 on the graph. Leaving the number of rollsat 1, click Roll dice three times. How many spots did each roll produce? What isthe average for the three rolls? You see that the graph displays at each point theaverage number of spots for all rolls up to the last one. Now you understand thedisplay.(b) Set the number of rolls to 100 and click Roll dice. The applet rolls the twodice 100 times. The graph shows how the average count of spots changes as wemake more rolls. That is, the graph shows x as we continue to roll the dice. Makea rough sketch of the final graph.(c) Repeat your work from (b). Click Reset to start over, then roll two dice 100times. Make a sketch of the final graph of the mean x against the number of rolls.Your two graphs will often look very different. What they have in common is thatthe average eventually gets close to the population mean = 7. The law of largenumbers says that this will always happen if you keep on rolling the dice.