### Long incubation period infects a population of people

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When a rare respiratory disease with a long incubation period infects a population of people, there is only a probability of 1/3 that an infected patient will show the symptoms within the ?rst month. When ?ve such symptomatic patients showed up in the only hospital in a small town, the astute doctor who treated these patients knew immediately that more will be coming in the next few months as the remaining infected members of the population begin to show symptoms. Assume that all symptomatic patients will eventually come to this one hospital.

(i) Postulate an appropriate probability model and use it to determine the "most likely" number of infected but not yet symptomatic patients, where x∗ is considered the most likely number if P (X = x∗) is the highest for all possible values of x.

(ii) Because of the nature of the disease, if a total of more than 15 people are infected, the small town will have to declare a state of emergency. What is the probability of this event happening?

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