Answer:
a)
<em> The probability that both will be hopelessly romantic is </em>
<em> P(X = 2) = 0.0361 </em>
<em>b) </em>
<em>The probability that at least one person is hopelessly romantic is</em>
<em> P( X>1) = 0.3439</em>
<u>Step-by-step explanation:</u>
a)
<em> Given data population proportion 'p' = 19% =0.19</em>
<em> q = 1-p = 1- 0.19 =0.81</em>
<em> Given two people are randomly selected</em>
<em> Given n = 2</em>
Let 'X' be the random variable in binomial distribution
![P(X=r) =n_{C_{r} } p^{r} q^{n-r}](https://tex.z-dn.net/?f=P%28X%3Dr%29%20%3Dn_%7BC_%7Br%7D%20%7D%20p%5E%7Br%7D%20q%5E%7Bn-r%7D)
<em>The probability that both will be hopelessly romantic is </em>
![P(X= 2) =2_{C_{2} } (0.19)^{2} (0.81)^{2-2}](https://tex.z-dn.net/?f=P%28X%3D%202%29%20%3D2_%7BC_%7B2%7D%20%7D%20%280.19%29%5E%7B2%7D%20%280.81%29%5E%7B2-2%7D)
P(X = 2) = 1 × 0.0361
<em> The probability that both will be hopelessly romantic is </em>
<em> P(X = 2) = 0.0361 </em>
<em>b) </em>
<em>The probability that at least one person is hopelessly romantic is</em>
<em> P( X>1) = 1-P(x<1)</em>
<em> = 1 - ( p(x =0)</em>
<em> = </em>
<em></em>
<em> </em> = 1 - (0.81)²
= 1 -0.6561
= 0.3439
<em>The probability that at least one person is hopelessly romantic is</em>
<em> P( X>1) = 0.3439</em>