Question

About 19​% 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​?

​(a) The probability that both will be hopelessly romantic is

0.0361.

​(Round to four decimal places as​ needed.)

​(b) The probability that at least one person is hopelessly romantic is

0.3439.

​(Round to four decimal places as​ needed.)

Answer 1

Answer:

a)

The probability that both will be hopelessly romantic is  

                                                             P(X = 2)   =  0.0361

b)

The probability that at least one person is hopelessly romantic is

               P( X>1) = 0.3439

Step-by-step explanation:

a)

Given data population proportion 'p' = 19% =0.19

                                                          q  = 1-p = 1- 0.19 =0.81

Given  two people are randomly​ selected

 Given n = 2

Let 'X' be the random variable in binomial distribution

$P(X=r) =n_{C_{r} } p^{r} q^{n-r}$

The probability that both will be hopelessly romantic is

                   $P(X= 2) =2_{C_{2} } (0.19)^{2} (0.81)^{2-2}$

                     P(X = 2)   = 1 × 0.0361

The probability that both will be hopelessly romantic is  

                                                             P(X = 2)   =  0.0361

b)

The probability that at least one person is hopelessly romantic is

               P( X>1) = 1-P(x<1)

                            = 1 - ( p(x =0)

                            = $1- 2_{C_{0} } (0.19)^{0} (0.81)^{2-0}$

                           = 1 - (0.81)²

                           = 1 -0.6561

                           = 0.3439

The probability that at least one person is hopelessly romantic is

               P( X>1) = 0.3439