We began this chapter by discussing Bayes' theorem. We learned that this theorem is used to revise prior probabilities to posterior probabilities, which are revised probabilities based on new information. We also saw how to use a probability revision table (and Bayes' theorem) to update probabilities in a decision problem. In Section 17.2 we presented an introduction to decision theory. We saw that a decision problem involves states of nature, alternatives, payoffs, and decision criteria, and we considered three degrees of uncertainty—certainty, uncertainty, and risk. In the case of certainty, we know which state of nature will actually occur. Here we simply choose the alternative that gives the best payoff. In the case of uncertainty, we have no information about the likelihood of the different states of nature. Here we discussed two commonly used decision criteria—the maximin criterion and the maximax criterion. In the case of risk, we are able to estimate the probability of occurrence for each state of nature. In this case we learned how to use the expected monetary value criterion. We also learned how to construct a decision tree in Section 17.2, and we saw how to use such a tree to analyze a decision problem. In Section 17.3 we learned how to make decisions by using posterior probabilities. We explained how to perform a posterior analysis to determine the best alternative for each of several sampling results. Then we showed how to carry out a preposterior analysis, which allows us to assess the worth of sample information. In particular, we saw how to obtain the expected value of sample information. This quantity is the expected gain from sampling, which tells us the maximum amount we should be willing to pay for sample information. We concluded this chapter with Section 17.4, which introduced using utility theory to help make decisions.