What does this equation mean?

Suppose an evaluation runs n independent trials of a consequential task and observes zero failures above the severity threshold $\tau$ . It is tempting to read that as evidence the risk is negligible. It is not, and the size of the gap has a name: the rule of three. If the true failure probability is $q_\tau$ , the probability of observing zero events in n independent trials is (1-$q_\tau$)^n ; setting that probability equal to 0.05 and solving gives n $\ln(1-q_\tau)$ $\approx$ -3 , and for small $q_\tau$ , where $\ln(1-q_\tau)$ $\approx$ -$q_\tau$ , this reduces to the boundary $q_\tau \lesssim \frac{3}{n}$. as the 95 percent upper confidence bound on the true rate, a result formalised by Hanley and Lippman-Hand’s 1983 paper on exactly this mistake — concluding, from a run of trials with no adverse events, that the underlying risk is zero [ 10 ] . Running 300 trials and seeing no catastrophic failures does not mean the failure rate is zero; it means the failure rate could plausibly be as high as one in a hundred and this experiment would still, quite often, have observed nothing.