Question

about 17% 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.)

Explanation:

Step1: Identify probability of one person

Let $p = 0.17$ be the probability that a person is hopelessly romantic.

Step2: Calculate probability of both being romantic

Since the events are independent, the probability that both of the two - selected people are hopelessly romantic is $P(\text{both})=p\times p$.
$P(\text{both}) = 0.17\times0.17= 0.0289$

Step3: Calculate probability of neither being romantic

The probability that a person is not hopelessly romantic is $q=1 - p=1 - 0.17 = 0.83$.
The probability that neither of the two - selected people is hopelessly romantic is $P(\text{neither})=q\times q=0.83\times0.83 = 0.6889$.

Step4: Calculate probability of at least one being romantic

The probability of at least one being romantic is $P(\text{at least one})=1 - P(\text{neither})$.
$P(\text{at least one})=1 - 0.6889 = 0.3111$

Answer:

(a) $0.0289$
(b) $0.3111$