Final answer:
In this problem, we are given that 25% of people in the United States say they would abandon their family forever for $1,000,000. We are asked to find the probability of various scenarios when choosing 5 people at random.
Step-by-step explanation:
In this problem, we are given that 25% of people in the United States say they would abandon their family forever for $1,000,000. We are asked to find the probability of various scenarios when choosing 5 people at random.
a. To find the probability that all 5 people would abandon their family, we need to multiply the probability of each person abandoning their family. The probability that one person would abandon their family is 25%. Therefore, the probability that all 5 people would abandon their family is (0.25)^5 = 0.00391 or 0.391%.
b. To find the probability that none of the 5 people would abandon their family, we need to find the complement of the probability that at least one person would abandon their family. The probability that at least one person would abandon their family is 1 - the probability that no one would abandon their family, which is (1 - 0.25)^5 = 0.2373 or 23.73%.
c. To find the probability that at least one person would abandon their family, we can find the complement of the probability that none of the 5 people would abandon their family. The probability that at least one person would abandon their family is 1 - the probability that no one would abandon their family, which is (1 - 0.25)^5 = 0.2373 or 23.73%.
d. To find the probability that exactly two people would abandon their family, we can use the binomial probability formula. The probability of exactly two successes in five trials is given by the 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, p is the probability of success, and C(n, k) is the number of combinations of n things taken k at a time. Here, n = 5, k = 2, and p = 0.25. Plugging in these values, we get P(X = 2) = C(5, 2) * 0.25^2 * (1 - 0.25)^(5 - 2) = 10 * 0.25^2 * 0.75^3 = 0.08789 or 8.789%.