Standard Statistical Concepts: Can They Produce Incoherence?

The present article concerns statistical concepts that are usually presented in the statistical classroom. Examples are presented in a way that simple applications of these concepts produce incoherent conclusions. The examples illustrate that: iid random variables are in fact strongly dependent; conditional probabilities may depend on how the conditioning arguments were learned; confidence intervals may have the property of diminished precision when information is increasing; and significance tests may not reject impossible hypotheses.