In June $2021,$ during the vaccine rollout for the Covid$-19$ emergency, it wasestimated that $90\%$ of the population over $50$ years old were fully vaccinated,while only $6\%$ were completely unvaccinated. $($The remaining $4\%$ had only onedose or had an unknown vaccination status, and therefore will not be consideredhere.$)$A Public Health England report on cases and hospitalisation from the $$delta$$variant $($originally sequenced in India$)$ was published at the end of June $2021.$The report showed that, between February and June $2021,$ among the $418$people admitted to the hospital with the $$delta$$ variant:$-163$ were fully vaccinated$-136$ were not vaccinated$-$ The remaining people had only one dose or an unknown vaccination status andwill not be considered hereOne may therefore wrongly conclude that a fully vaccinated person is surprisinglymore likely to be hospitalised than an unvaccinated person. Using Bayes$$ theorem tocalculate the relevant probabilities from the data above, prove that this claim iswrong. Show that this data actually proves that vaccines are extremely effective atreducing the risk of hospitalisation after contracting the $$delta$$ variant.