Example — Lethality of a virus

Example — Lethality of a virus

During an epidemic, a virus $A$ strikes about $0{,}1\%$ of the population: $P(A) = 0,001$. The reported lethality (the probability of dying once the virus has been contracted) is $P(B\mid A) = 0,03$, where $B$ = “the subject dies”.

Question: what is the overall probability of dying because of the virus? By the law of compound probability:

$P(A\cap B) = P(A)\cdot P(B\mid A) = 0,001\cdot 0,03 = \boxed{0,00003}.$

Three people in a hundred thousand: a figure that, multiplied by millions of inhabitants, nonetheless has a considerable impact.

Reverse question: among those who died in the period, knowing that in total $P(B) = 0,00037$, what fraction is attributable to the virus?

$P(A\mid B) = \frac{P(A\cap B)}{P(B)} = \frac{0,00003}{0,00037} \approx \boxed{0,081 \approx 8\%}.$

So about $8\%$ of the overall deaths is attributable to the virus.

Caution

$P(A\mid B)$ and $P(B\mid A)$ are very different quantities: confusing them is a classic mistake. Here the lethality $P(B\mid A)=3\%$ and the fraction of deaths attributable to the virus $P(A\mid B)\approx 8\%$ answer two completely distinct questions.

Links

Topics: Probability
Concepts: Product formula · Conditional probability · Bayes’ theorem
Methods: Bayes · Conditional probability
Skills: Probability calculation