Final answer:
To find the probabilities, we will use the binomial probability formula and substitute the values of n, k, and p. For each scenario, we calculate the probabilities for the given range of k values and sum them up.
Step-by-step explanation:
To find the probability, we will use the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, n = 30, p = 0.8, and we need to calculate the probabilities for different values of k.
a. To find the probability that exactly 23 of them live in cities with population greater than 100,000 people, we can substitute n=30, k=23, and p=0.8 into the formula: P(X=23) = C(30, 23) * 0.8^23 * (1-0.8)^(30-23).
b. To find the probability that at most 25 of them live in cities with population greater than 100,000 people, we need to calculate the probabilities for k=0, 1, 2, ..., 25 and sum them up.
c. To find the probability that at least 24 of them live in cities with population greater than 100,000 people, we need to calculate the probabilities for k=24, 25, ..., 30 and sum them up.
d. To find the probability that between 22 and 26 (including 22 and 26) of them live in cities with population greater than 100,000 people, we need to calculate the probabilities for k=22, 23, ..., 26 and sum them up.