About \( 12 \% \) of the population of a large country is hopelessly romantic. If two people are randomly selected, what is the probability both are hopelessly romantic? What is the probability at least one is hopelessly romantic? Assume the events are independent. (a) The probability that both will be hopelessly romantic is 0.0144 . (Round to four decimal places as needed.) (b) The probability that at least one person is hopelessly romantic is (Round to four decimal places as needed.)

To calculate the probabilities, let's first clarify the probabilities concerning the "hopelessly romantic" population. Given that \( P(R) = 0.12 \) (the probability of being hopelessly romantic), the probability of not being hopelessly romantic is \( P(NR) = 1 - P(R) = 0.88 \). (a) The probability that both selected individuals are hopelessly romantic can be calculated as follows: \[ P(\text{both romantic}) = P(R) \times P(R) = 0.12 \times 0.12 = 0.0144. \] So, you got that one right! (b) To find the probability that at least one person is hopelessly romantic, we can use the complement rule: \[ P(\text{at least one romantic}) = 1 - P(\text{both not romantic}) = 1 - P(NR) \times P(NR). \] \[ P(\text{both not romantic}) = 0.88 \times 0.88 = 0.7744. \] Now, calculate: \[ P(\text{at least one romantic}) = 1 - 0.7744 = 0.2256. \] Therefore, the probability that at least one of the two selected individuals is hopelessly romantic is \( 0.2256 \) (rounded to four decimal places as needed).