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If the body produces random antibodies in order to fight a virus, is there a small chance that the specific antibody needed to fight the virus just won't be randomly generated? Even though it is a common virus that normally doesn't kill people?

For example if you have a common cold, is there a small chance that you could die from this just by probability? Whereas someone else won't because they randomly produced the right antibody?

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Yes and no. Yes, it's possible that some individuals won't be able to (or happen to) make antibodies to a specific part of a pathogen. But most pathogens are large and complex enough that they have many different targets for antibodies, and it becomes increasingly unlikely that someone won't be able to target any part of the pathogen.

An example is given in Influence of Immune Priming and Egg Adaptation in the Vaccine on Antibody Responses to Circulating A(H1N1)pdm09 Viruses After Influenza Vaccination in Adults. A subset of adults (who were exposed to a particular influenza virus as children) have poor antibody responses to a vaccine to a different strain of influenza virus:

Significantly reduced antibody reactivity to circulating viruses bearing K163Q was observed only in the USSR/77-primed cohort, whereas significantly lower reactivity caused by egg-adapted Q223R change was detected across all 3 cohorts.

(This is not an argument against vaccination, I feel obliged to point out, because the vaccinated people were still better off than unvaccinated, just not as well protected as they could be.)

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