This chapter provides basic terminology and methods for the discussion of Bonferroni-type inequalities. As illustrated in Chapter 1, Bonferroni-type inequalities play a critical role in various applied fields including medicine, finance, education, economics, and engineering. The main principle underpinning the applications of probability inequalities in Chapter 1 is the evaluation of the joint probability by means of marginal distributions (or joint probabilities with reduced dimensions). For example, in a dose-response study, when the experimenter is interested in constructing a simultaneous confidence set that estimates differences of drug effects among several treatments with a control, the joint probability of confidence intervals for the comparison between the ith dose effect and the placebo, Ai, for i = 1, ...,k, reads

Ai) = 1−P( k⋃

Evaluation of the simultaneous confidence level on the left-hand side is converted into the evaluation of the probability of the union of a set of events on the right-hand side.