The following statistics give hospitalisation rates for COVID cases in an outbreak of the delta variant in Auckland in 2021.

The table shows all COVID cases aged 12 years and over.

a. The sample space is Ω = {Auckland residents aged 12+}.
Define events C and H for respectively contracting COVID and being hospitalised; and events N, P, and F for the three vaccine statuses. Use the table to write down the conditional probability of hospitalisation for COVID patients with each of the three vaccine statuses.

b. In the Auckland outbreak, the probability of contracting COVID for unvaccinated people was 10 times that for fully vaccinated people; and for partially vaccinated people it was 6 times that for the fully vaccinated. Write down this information in probability statements.

c. Define KΩ = P(C | F)
P(C)
.
Using your answer to (b), write down expressions for P(N | C), P(P | C), and P(F | C) in terms of KΩ.

d. Another NZ city is preparing for its own delta outbreak.
Their 12+ population is 82% fully vaccinated, 3% partially vaccinated, and 15% unvaccinated. Estimate what casehospitalisation rate you think they should expect. (The case-hospitalisation rate is the probability of hospitalisation for each COVID case.) Explain your reasoning.