\( 61 \% \) 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.

To solve these problems, we can model the number of people living in cities with populations greater than 100,000 using the binomial distribution, where \( n \) is the total number of people surveyed, \( p = 0.61 \) is the probability of living in a large city, and \( k \) is the number of people we're interested in. Assuming \( n = 30 \) for this example, we can calculate: a. The probability of exactly 20 people living in such cities can be calculated using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substituting \( n = 30 \) and \( k = 20 \) gives: \[ P(X = 20) = \binom{30}{20} (0.61)^{20} (0.39)^{10} \] b. The probability of at most 20 can be found by summing the probabilities from 0 to 20 using: \[ P(X \leq 20) = \sum_{k=0}^{20} P(X = k) \] c. The probability of at least 19 can be calculated similarly: \[ P(X \geq 19) = \sum_{k=19}^{30} P(X = k) \] d. Finally, for the probability between 16 and 20, we sum the probabilities from 16 to 20: \[ P(16 \leq X \leq 20) = \sum_{k=16}^{20} P(X = k) \] Calculating these probabilities would typically require either a binomial table or computational software. By plugging in the numbers or using software, you can find the actual probability values for each scenario! Happy calculating!