1111 Use the approach given on the previous page for the following. (a) Find the probabilities of having 0, 1, 2, or 3 boys in a family of 3 children. (b) Find the probabilities of having 0, 1, 2, 3, 4, 5, or 6 girls in a family of 6 children. 43. (Modeling) Spread of Disease 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 P = a S kbqk11 - q2S-k, where q = 11 - p2I, 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. (Data from Hoppensteadt, F., and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) 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. (Hint: 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. 44. (Modeling) Spread of Disease (Refer to Exercise 43.) Suppose that in a family I = 2 and S = 4. If the probability P is 0.25 of there being k = 2 uninfected members after 1 week, find the possible values of p to the nearest thousandth. (Hint: Write P as a function of p.) Chapter 11Test Prep Key Terms 11.1 finite sequence infinite sequence terms (of a sequence) general term (nth term) convergent sequence divergent sequence recursive definition Fibonacci sequence series summation notation finite series infinite series index of summation 11.2 arithmetic sequence (arithmetic progression) common difference arithmetic series 11.3 geometric sequence (geometric progression) common ratio geometric series annuity future value (of an annuity) 11.4 Pascal’s triangle factorial notation binomial coefficient binomial theorem (general binomial expansion) 11.6 tree diagram independent events permutation combination 11.7 trial outcome sample space event probability certain event impossible event complement Venn diagram odds compound event mutually exclusive events binomial experiment CHAPTER 11 Test Prep
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