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 (Round to four decimal places as needed.)

To find the probability that both selected people are hopelessly romantic, we can use the formula for independent events: \[ P(A \text{ and } B) = P(A) \times P(B) \] Here, the probability (\(P\)) of a person being hopelessly romantic is \(0.12\) (or \(12\%\)). Therefore, the probability that both are hopelessly romantic is: \[ P(\text{both are romantic}) = 0.12 \times 0.12 = 0.0144 \] So, rounding to four decimal places, the probability that both will be hopelessly romantic is \(0.0144\). Now, to find the probability that at least one person is hopelessly romantic, we can use the complement rule: \[ P(\text{at least one}) = 1 - P(\text{neither}) \] First, we calculate the probability that neither person is hopelessly romantic: \[ P(\text{not romantic}) = 1 - 0.12 = 0.88 \] The probability that both are not romantic is: \[ P(\text{neither}) = 0.88 \times 0.88 = 0.7744 \] Then we can find the probability that at least one is hopelessly romantic: \[ P(\text{at least one}) = 1 - 0.7744 = 0.2256 \] Thus, the probability that at least one is hopelessly romantic is \(0.2256\).