Separate infection counts from a later conditional-test model

During a pandemic, an unknown infectious disease spread through a population. Researchers wanted to study the probability, $p$, that a person is infected. They collected some data from a random sample of 25 people.

(i) State, in the context of this question, two assumptions needed for the number of infected people to be well modelled by a binomial distribution.2 marks

Assume now that these assumptions do in fact hold.

(ii) It is given that the probability that at most 2 people are infected is 0.0982. Write an equation in $p$ and solve for $p$.2 marks

A total of 30 random samples of 25 people were chosen from this population.

(iii) Find the probability that the total number of samples with at most 2 infected people is at least 5.3 marks

It is now given that $p=0.10$.

A diagnostic test for the presence of the disease was carried out on the population. It was found that the percentage of infected people who were tested positive is 85%, while the percentage of non-infected people who were tested negative is 96%.

(iv) Find the probability that a person who tested positive carries the disease.3 marks

(v) Hence or otherwise, find the probability that a person who tested positive does not carry the disease.1 mark

(vi) Discuss briefly if the diagnostic test is worthwhile.1 mark