Testing Hypotheses in Particle Physics: Plots of $p_0$ Versus $p_1$

For situations where we are trying to decide which of two hypotheses $H_0$ and $H_1$ provides a better description of some data, we discuss the usefulness of plots of $p_0$ versus $p_1$, where $p_i$ is the $p$-value for testing $H_i$. They provide an interesting way of understanding the difference between the standard way of excluding $H_1$ and the $C{L}_s$ approach; the Punzi definition of sensitivity; the relationship between $p$-values and likelihood ratios; and the probability of observing misleading evidence. They also help illustrate the Law of the Iterated Logarithm and the Jeffreys-Lindley paradox.