How can I determine the efficiency of a vaccine in the following situation? In a country of 20 mil. people, 25% of the inhabitants are fully vaccinated anti Covid-19 and 75% not vaccinated at all. It was noticed that, in the last weeks, 82% of the people found positive, after being tested, never received the vaccine and only 9% of those who died due of Covid-19 had received the vaccine.
In short:

*

*25% vaccinated,

*75% not-vaccinated,

*82% of the infected subjects are not-vaccinated,

*9% of the deceased people are vaccinated.

Question: What are the two efficiencies of the vaccine: (1) regarding its power to prevent an infection, (2) concerning its ability to prevent death.
If is self evident that, had the vaccine had no effect (had it been just a placebo) 75% of the people who would have died, and 75% of those infected, would have not been vaccinated and the rest of 25% would have been vaccinated. In this case the efficiencies of the vaccine would have been zero.
It is also evident that if the only people who died or got infected had been not-vaccinated, than the efficiencies of the vaccine would have been both 100%.
From this point forward, I don't really know how to continue the reasoning for finding the two efficiencies.
(Remark: You can make any simplifying assumption you would like.)
 A: Here are some ideas to get started. Following the notation of Wikipedia, let the (unknown) attack rates of the infection be $ARU$ and $ARV$ in the vaccinated and unvaccinated populations respectively. (What follows is directed to part (1), but the idea applies to part (2) as well.) For convenience, I'll use the notation of the Wikipedia article I linked in comments to the OP.
Since the efficiency may be written as $$\text{efficiency} = \frac{ARU-ARV}{ARU} = 1-\frac{ARV}{ARU},$$ it suffices to compute the ratio of these attack rates. Next, to make the computation a bit more concrete we may focus on a representative sample---say, 400 persons. (This specific number doesn't actually matter but it's convenient enough.) Of these 400, 100 are vaccinated and the remaining 300 are not. From this and the definition of attack rate, we may express symbolically:

*

*how many of the 100 vaccinated persons will be infected

*how many of the 300 unvaccinated persons will be infected

*how many people out of the 400 are infected in total

From these, we can express (symbolically) what fraction of those infected in this 400-person sample did or did not receive the vaccine. This will give a condition on the attack rates, and from this there is enough info to find the efficiency.
I will note that the above is not the most direct calculation possible. (Use of Bayes' theorem reduces it to one line, for instance.) But for now I wanted to provide a more intuitive starting point.
