State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution.
Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.
Find the mean and variance of the number of people with the disease in a random sample of 100 people.
A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.