1. Pennsylvania lottery. In 1980, millions of viewers watched as the number 666 was drawn for the Pick 3 Pennsylvania lottery. Although as likely as any other num- ber, lottery authorities and local bookmakers became suspi- cious when they noticed that a large number of tickets were purchased for the eight numbers 444, 446, 464, 466, 644, 646, 664, and 666. Soon, a handful of players came forward to claim approximately $1.8 million of the then-record 3.5 million payout. Officials eventually discovered that 8 of the 10 balls were weighted, giving them almost no chance to be selected in the drawing. Is this an example of special- or common-cause variation? Explain.

2. Out-of-control rules. Most control charts use three sig- mas (standard deviations) to set the upper and lower control limits. For a run chart, what is the probability that an observation would fall beyond 3s from the mean based on random chance variation alone? What assumptions do you have to make to answer this?

3. Out-of-control rules, part 2. Instead of one observation outside of 3s from the mean as in Exercise 17, suppose a quality control engineer decides to use only the run chart rule that the system is out of control if 8 observations in a row lie on the same side of the mean. What is the probability that would happen based on random chance variation alone? What assumptions do you have to make to answer this?