Solve the problem.
What will happen when an infectious disease is introduced into a family? Suppose a family has I infected members and S members who are not infected but are susceptible to contracting the disease. The probability P of exactly k people not contracting the disease during a 1-week period can be calculated by the formula
where q = (1 – p)I, and p is the probability that a susceptible person contracts the disease from an infected person. For example, if p = 0.5, then there is a 50% chance that a susceptible person exposed to 1 infected person for 1 week will contract the disease. Give all answers to the nearest thousandth.
(a) Compute the probability P of 3 family members not becoming infected within 1 week if there are currently 2 infected and 4 susceptible members. Assume that p = 0.1. To use the formula, first determine the values of k, I, S, and q.
(b) A highly infectious disease can have p = 0.5. Repeat part (a) with this value of p.
(c) Determine the probability that everyone will become sick in a large family if, initially, I = 1, S = 9, and p = 0.5.
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