\( 61 \% \) of all Americans live in cities with population greater than 100,000 people. It randomly selected, find the probability that a. Exactly 20 of them live in cities with population greater than 100,000 people. b. At most 20 of them live in cities with population greater than 100,000 people. c. At least 19 of them live in cities with population greater than 100,000 people. d. Between 16 and 20 (including 16 and 20) of them live in cities with population greater than 100,000 people.

We are told that 61% of Americans live in cities with a population greater than 100,000. Suppose that a random sample of 30 Americans is selected. (The sample size is not stated explicitly in the problem text, but the numbers used in parts (a)–(d) strongly suggest that n = 30.) Let   X = the number of people in the sample who live in a city of over 100,000 people. Since each person is independent and the probability that an individual lives in such a city is p = 0.61, we have   X ~ Binomial(n = 30, p = 0.61). Using the binomial probability formula   P(X = k) = (n choose k) · (p)^k · (1 – p)^(n – k), we can answer the questions as follows: ────────────────────────────── (a) Exactly 20 of them live in cities with population greater than 100,000. This is just the probability that X = 20:   P(X = 20) = C(30,20) · (0.61)^20 · (0.39)^(10). ────────────────────────────── (b) At most 20 of them live in cities with population greater than 100,000. “At most 20” means X ≤ 20, so we add up the probabilities for k = 0, 1, 2, …, 20:   P(X ≤ 20) = Σ (from k = 0 to 20) [C(30,k) · (0.61)^k · (0.39)^(30 – k)]. ────────────────────────────── (c) At least 19 of them live in cities with population greater than 100,000. “At least 19” means X ≥ 19, so we add up the probabilities for k = 19, 20, …, 30:   P(X ≥ 19) = Σ (from k = 19 to 30) [C(30,k) · (0.61)^k · (0.39)^(30 – k)]. ────────────────────────────── (d) Between 16 and 20 (including 16 and 20) of them live in cities with population greater than 100,000. This is the sum for k = 16, 17, 18, 19, 20:   P(16 ≤ X ≤ 20) = Σ (from k = 16 to 20) [C(30,k) · (0.61)^k · (0.39)^(30 – k)]. ────────────────────────────── Any answer that correctly expresses these probabilities (or their numerical approximations) is acceptable. For example, one acceptable answer is:   (a) P(X = 20) = (30 choose 20)·(0.61)^20·(0.39)^10   (b) P(X ≤ 20) = Σₖ₌₀²⁰ (30 choose k)·(0.61)^k·(0.39)^(30–k)   (c) P(X ≥ 19) = Σₖ₌₁⁹⁰ (30 choose k)·(0.61)^k·(0.39)^(30–k)   (d) P(16 ≤ X ≤ 20) = Σₖ₌₁₆²⁰ (30 choose k)·(0.61)^k·(0.39)^(30–k). (Any algebraically equivalent answer is correct.)