COVID treatments

COVID (abbreviated by $C$) has similar symptoms to a certain type of flu (let us call it $A$). At some point, we know that in a population of students, 20% have COVID, 50% have Flu $A$, and the rest have other diseases (everyone in this group is sick! but with exactly one illness). Let us consider treatment $T$. We know that 80% of COVID patients respond to treatment $T$, while only 10% of patients with Flu A respond to this treatment. As for the rest of this group of students, 30% respond to the treatment.

What is the probability that a patient (from this group) responds to treatment $T$? In this case, let us denote the sample space by $\Omega$ containing all the students in this population of interest. We are interested in ${\mathbb P}[T]$ where $T \subseteq \Omega$ contains the patients that respond to the treatment.

${\mathbb P}[T]= {\mathbb P}[T \mid C] \cdot {\mathbb P}[C] + {\mathbb P}[T \mid A] \cdot {\mathbb P}[A] + {\mathbb P}[T \mid R] \cdot {\mathbb P}[R]$

Plugging in the appropriate numbers gives us the result.

Next, suppose the government does not want to pay the cost of the (expensive) test that distinguishes between different types of diseases. In this case, the doctor tries treatment $T$ and finds that the patient (luckily) responds to the treatment. What is the probability that the patient had COVID? In this case, we are looking for ${\mathbb P}[C \mid T]$ which is equal to

$\frac{{\mathbb P}[C\cap T]}{{\mathbb P}[T]}=\frac{{\mathbb P}[T \mid C] \cdot {\mathbb P}[C]}{{\mathbb P}[T]}$

We know all these values, so we should be able to compute the result 😊