This work is devoted to the study of the probability of immunity, i.e. the effect occurs whether exposed or not. We derive necessary and sufficient conditions for non-immunity and $\epsilon$-bounded immunity, i.e. the probability of immunity is zero and $\epsilon$-bounded, respectively. The former allows us to estimate the probability of benefit (i.e., the effect occurs if and only if exposed) from a randomized controlled trial, and the latter allows us to produce bounds of the probability of benefit that are tighter than the existing ones. We also introduce the concept of indirect immunity (i.e., through a mediator) and repeat our previous analysis for it. Finally, we propose a method for sensitivity analysis of the probability of immunity under unmeasured confounding.

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