Respuesta :
Answer:
a) There is a 92% probability that a randomly selected woman in that age group is married or is participating in the labor force.
b) There is a 31% probability that a randomly selected woman in that age group is married or is participating in the labor force but not both.
c) This means that there is an 8% probability that a randomly selected woman in that age group is neither married nor participating in the labor force.
Step-by-step explanation:
We can solve this problem building the Venn's diagram of these probabilities.
I am going to say that
The set A are those women who participate in the work force.
The set B are those women who are married.
We have that:
[tex]A = a + (A \cap B)[/tex]
In which a represents those that participate in the work force but are not married and [tex](A \cap B)[/tex] are those who are both on the work force and married.
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
We start finding the values from the intersection of these sets:
Suppose also that 61% of women 25 through 49 years of age are married and are participating in the labor force.
This means that [tex]A \cap B = 0.61[/tex]
Suppose 78% of the women in that age group are married.
This means that [tex]B = 0.78[/tex]
[tex]B = b + (A \cap B)[/tex]
[tex]0.78 = b + 0.61[/tex]
[tex]b = 0.17[/tex]
75% of the women 25 through 49 years of age participate in the labor force.
This means that [tex]A = 0.75[/tex]
[tex]A = a + (A \cap B)[/tex]
[tex]0.75 = a + 0.61[/tex]
[tex]a = 0.14[/tex]
a. What is the probability that a randomly selected woman in that age group is married or is participating in the labor force
This is
[tex]P = a + b + (A \cap B) = 0.14 + 0.17 + 0.61 = 0.92[/tex]
There is a 92% probability that a randomly selected woman in that age group is married or is participating in the labor force.
b. What is the probability that a randomly selected woman in that age group is married or is participating in the labor force but not both?
This is
[tex]P = a + b[/tex]
[tex]P = 0.14 + 0.17[/tex]
[tex]P = 0.31[/tex]
There is a 31% probability that a randomly selected woman in that age group is married or is participating in the labor force but not both.
c. What is the probability that a randomly selected woman in that age group is neither married nor participating in the labor force?
From a), we found that there is a 92% probability that a randomly selected woman in that age group is married or is participating in the labor force.
This means that there is an 8% probability that a randomly selected woman in that age group is neither married nor participating in the labor force.
