# Chapter 1. The Law of Large Numbers

## Introduction

Statistical Applets

Choose the number of dice by clicking FEWER DICE or MORE DICE. To see the roll totals or the population mean number of spots μX, check the corresponding box at the left. Fill in the number of rolls you'd like to make and click ROLL DICE. The graph shows the average x̄ of the spots for all of your rolls. As you make more and more rolls, x̄ should eventually get close to μX.

Click the "Quiz Me" button to complete the activity.

Observe a random variable X very many times. In the long run, the proportion of outcomes taking any value gets close to the probability of that value. The Law of Large Numbers says that the average of the observed values gets close to the mean μX of X.

In this applet, we represent a random variable X as the total number of spots on the "up" faces of one or more dice.

### Question 1.1

Set the number of dice to 1.

• In the long run, the average number of spots µX that should come up when you roll one die is brVrJl3J7nE=.
• Roll the die 5 times. What is the average number of spots observed over these 5 rolls? whvjMhXDKBkzIL8g.
• Now roll the die 45 more times, for a total of 50 rolls. What is the average number of spots observed over these 50 rolls? whvjMhXDKBkzIL8g.
3
µX is the value shown by the blue line when you click "Show µX". Try again.
Incorrect. See above for the correct answers.
Great job.

### Question 1.2

Now set the number of dice to 6.

• In the long run, the average number of spots µX that should come up when you roll six dice is Sh66xq0desA=.
• Roll the six dice 5 times. What is the average number of spots observed over these 5 rolls? LfKwD6W+290=.
• Now roll the dice 45 more times, for a total of 50 rolls. What is the average number of spots observed over these 50 rolls? LfKwD6W+290=.
3
µX is the value shown by the blue line when you click "Show µX". Try again.
Incorrect. See above for the correct answers.
Great job.

### Question 1.3

vm9QxJyeasdbav17OZdEMK24ikFyxh1N6WFwhQYy0QPpSw7JmfBqtpyH1SmWwV/aNrowb84+vO+i2E8rypsmuIhnCubsmnf3MOcV909sVu5NzfwKVge8PQ6aILiL5/TR9hhJNoDmaTKMBuJ/LRtim3M11mk/qybklHncfGNkw56wzNHra7O6zwtus929qmihDfaJoe2XT6+XU479n8ymrLsmWv651fnjurOP7T81WcolSPH3sClYFhaugatFJbuP+nnhN4Olbcuf3+bviJaiRLljQHg3AJNJ
The number of spots on any one roll is highly variable. However, the law of large numbers says that the more rolls observed, the closer the average roll should get to µX. Therefore, the observed average will usually be closer to µX after 50 rolls than after 5 rolls. (However, since there is a lot of randomness involved here, once in a while the law of large numbers will be "mistaken", and the average after 50 rolls will actually be farther away from µX than the average after 5 rolls.)