What does 99.9% mean?

99.9% sounds like a very high proportion, easily interpretted as certainty in everyday life. But what does it mean when it really matters?

Availability

When someone claims 99.9% availability of their service, it means approximately 40 minutes unavailability per month.

Survival rate

Survival rate 99.9% in a single period means the probability of survival is 1/2 after ~700 periods.

Test specificity

Test specificity is the probability that the test result is negative when the queried situation is absent or $P(N|\neg I)$ (where $I$ stands for infection here, $N$ is negative result).

Test sensitivity is the $P(P|I)$, i.e. positive result given an infected sample.

\[ \begin{align} P(I|P) &= \frac{P(P|I) P(I)}{P(P)} \cr &= \frac{P(P|I) P(I)}{P(P|I)P(I) + P(P|\neg I)P(\neg I)} \cr &= \frac{P(P|I) P(I)}{P(P|I)P(I) + P(P|\neg I)(1 - P(I))} \cr &= \frac{P(P|I) P(I)}{P(P|I)P(I) + (1-P(N|\neg I))(1 - P(I))} \cr &= \frac{\text{sens} \cdot P(I)}{\text{sens} \cdot P(I) + (1-\text{spec})(1 - P(I))} \cr &= \frac{\text{sens}}{\text{sens} + (1-\text{spec})\left(\frac1{P(I)} - 1\right)} \end{align} \]

Let’s have prevalence $P(I) = 0.1\%$, moderately sensitive test ($\text{sens}=80\%$) and $\text{spec} = 99.9\%$. We have no big expectation about $P(\neg I|N)$ given the low sensitivity, however, we expect $P(I|P)$ to be high with such a high specificity. If we plug the numbers into the formula we get $P(I|P) \approx 50\%$. It looks like a coin flip but such a “test” would provide no new information, the coin gives $P(I|P) = P(I)$.