In this chapter we address the following problem.

Problem:   We have data being generated by some probabilistic phenomenon.   What can we reasonably infer about this phenomenon from our data, and how reliable are those inferences?

Here are some examples for consideration.

Example:   We see a sequence of   $1$s   and   $0$s   believed to be representing heads and tails of tosses of a fair coin.   We cannot determine if the coin is British or Canadian or Peruvian, but can we determine with some confidence whether or not the coin is indeed fair?   (Such an example was already seen in an earlier section.)

Example:   A region of Pakistan has a seemingly unusual preponderance of male babies.   Does this kind of variation in sexual birthrates happen frequently, or might there be some other issues to which we should be paying attention?

Example:   Before phasing in a new treatment, physicians want to know if it is demonstrably more effective than the current treatment.   If a study shows that thirty seven of one hundred people demonstrably improve with the current treatment, and forty three of one hundred improve with an experimental treatment, can we be fairly confident that a switch in treatment will produce improved outcomes?

Clearly there are many applications of such analyses to decision making.   Consider the possibilities in gaming, business (think about marketing and advertising strategies, for example), medicine and pharmacology, sports (should a lefty or righty pitcher be used against a given line-up), and many other fields.