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1. A company uses three different assembly lines A1, A2, and A3 to manufacture a particular component. Of those manufactured by line A1, 5% need re- work to remedy a defect, whereas 8% of A2 s components need rework and 10% of A3 s need re- work. Suppose that 50% of all components are pro- duced by line A1, 30% are produced by line A2, and 20% come from line A3. If a randomly selected com- ponent needs rework, what is the probability that it came from line A1? From line A2? From line A3?

2. Disregarding the possibility of a February 29 birth- day, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days.

a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same birthday?

b. With k replacing ten in part (a), what is the small- est k for which there is at least a 50-50 chance that two or more people will have the same birthday?

c. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [Note: The article Methods for Studying Coinci- dences (F. Mosteller and P. Diaconis, J. Amer. Statist. Assoc., 1989: 853-861) discusses prob- lems of this type

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