EXAMPLE 1 Are these dice loaded?

Suppose you are playing a dice game, where you roll a pair of dice and win the sum of the two dice in dollars. A fair price to pay to play this game is $7 a throw, because the long-run mean when tossing two fair dice is 7. Now suppose you have played this game 10 times (paying a total of $70), with the following 10 results from throwing the two dice:

4 6 2 7 8 3 5 4 9 2

These 10 dice rolls add up to 50, meaning that, for your outlay of $70, you have only received $50 in return. You wonder:

  • Are these dice fair but you have just had a streak of bad luck, or
  • Are these dice not fair, that is, loaded (weighted) to provide low outcomes?

This is a basic example of hypothesis testing, where we have two competing ideas, and we turn to observed data (the dice rolls) to provide evidence in favor of one idea or the other.

We examine this question in more detail in the exercises and again in Section 9.2.