Vaccines & Placebo/Nocebo

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With all the "Do not dare to talk bad about X" and general censorship, I started to wonder about Placebo and Nocebo effects on the mRNA vaccines, i.e.: As long as I believe it works the way it is intended to, it will work the way it is intended to.
I was looking up the study "Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine" [ https://www.nejm.org/doi/full/10.1056/nejmoa2034577 ], but being a brainlet I require help understanding a small part of the chapter Methods - Efficacy.
> Among 36523 participants who had no evidence of existing or prior SARS-CoV-2 infection, 8 cases of Covid-19 with onset at least 7 days after the second dose were observed among vaccine recipients and 162 among placebo recipients.
How is it that the numbers for the whole placebo-group (18325) are so low (162 of 18325 => 0.8840%), considering NCoVs notorious infectivity? Is it known if everybody actually had contact with the occuring NCoV?
On a sidenote in favour of recovered people:
> Among participants with and those without evidence of prior SARS CoV-2 infection, 9 cases of Covid-19 at least 7 days after the second dose were observed among vaccine recipients and 169 among placebo recipients, corresponding to 94.6% vaccine efficacy (95% CI, 89.9 to 97.3).
Assuming the deltas of prior infected (Nv=19965-18198=1767 and Np=20172-18325=1847), the prior infected+vaccinated (Vi=9-8=1), the prior infected+placeboed (Pi=169-162=7), corresponding percentage of failed protection as X people got COVID again (VEvi=Nv/Vi=1767/1=>0.06% / VEpi=Np/Pi=1847/7=>0.38%) and contra-probability of people not getting sick again (EFvi=100%-0.06%=99.94% / EFpi=100%-0.38%=99.62%) for vaccinated/placebo'd with pre-infection, shouldn't recovered people be at least as well-regarded as vaccinated people?

Thanks for any clarifications/corrections.