Respuesta :
Answer:
There is a 69.8% probability that a subscriber rented a car during the past 12 months for business or personal reasons.
There is a 30.2% probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons.
Step-by-step explanation:
We can solve this problem by building the "Venn Diagram" of these probabilities.
I am going to say that A is the probability that a magazine subscriber rented a car for business reasons.
B is the probability that a magazine subscriber rented a car for personal reasons.
C are those who did not rent a car for either of these reasons.
We have that:
[tex]A = a + (A \cap B)[/tex]
In which a are those who only rented for business reasons and [tex]A \cap B[/tex] are those who rented both for business and personal reasons.
By the same logic, we have that
[tex]B = b + (A \cap B)[/tex]
In which b are those who only rented for personal reasons.
The sum of the probabilities is 1, so:
[tex]a + b + (A \cap B) + C = 1[/tex]
We start finding the values from the intersection of these sets.
30% rented a car during the past 12 months for both business and personal reasons. So [tex]A \cap B = 0.3[/tex].
45.8% rented a car during the past 12 months for business reasons
This means that [tex]A = 0.458[/tex].
And
[tex]A = a + (A \cap B)[/tex]
[tex]0.458 = a + 0.3[/tex]
[tex]a = 0.158[/tex]
54% rented a car during the past 12 months for personal reasons
[tex]B = b + (A \cap B)[/tex]
This means that [tex]B = 0.54[/tex].
And
[tex]B = b + (A \cap B)[/tex]
[tex]0.54 = b + 0.3[/tex]
[tex]b = 0.24[/tex]
What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?
That is the probability that someone rented a car for only one of these reasons, or both. So:
[tex]P = a + b + (A \cap B) = 0.158 + 0.24 + 0.30 = 0.698[/tex]
There is a 69.8% probability that a subscriber rented a car during the past 12 months for business or personal reasons.
What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?
This is the value of C
We have that:
[tex]a + b + (A \cap B) + C = 1[/tex]
[tex]0.698 + C = 1[/tex]
[tex]C = 0.302[tex]
There is a 30.2% probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons.
