Answer to Question #235862 in Statistics and Probability for OPPONG

Question

survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 54% rented a car during the past

12 months for personal reasons, and 30% rented a car during the past 12 months for both business and personal reasons.

i. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?

ii. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons

Answer 1

QUESTION

survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 54% rented a car during the past 12 months for personal reasons, and 30% rented a car during the past 12 months for both business and personal reasons.

i. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?

ii. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?

SOLUTION

We can solve this problem by building the "Venn Diagram" of these probabilities.

I am going to assume that A is the probability that a magazine subscriber rented a car for business reasons and B is the probability that a magazine subscriber rented a car for personal reasons and C are those who did not rent a car for either of these reasons.

So we would have: $A=a + (A\bigcap B)$

Where:

$a$ represent those who only rented for business reasons.

$(A\bigcap B)$ represent those who rented both for business and personal reasons.

From the same logic, we have that: $B = b + (A \bigcap B)$, Where:

$b$ represent those who only rented for personal reasons.

The sum of the probabilities is 1, so:

$a+b+(A\bigcap B)+C=1$

Begin getting the values from the intersection of these sets.

30% of the people rented a car during the past 12 months for both business and personal reasons. So, $(A\bigcap B)=0.3$

45.8% rented a car during the past 12 months for business reasons, meaning that $A=0.458$

and $A=a+(A\bigcap B)$

$0.458=a+0.3$

$a=0.158$

54% rented a car during the past 12 months for personal reasons.

$B=b+(A\bigcap B)$ Meaning that $B=0.54$

And $B=b+(A\bigcap B)$

$0.54=b+0.3$

$b=0.21$

SO WE MOVE TO SOLVING THE QUESTIONS:

i. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?

Solution:

That is the probability that someone rented a car for only one of these reasons, or both.

So: $P=a+b+(A\bigcap B)$

$P=0.158+0.24+0.3$

$P=0.698$

69.8% probability that a subscriber rented a car during the past 12 months for business or personal reasons.

ANSWER: $69.8\%$

ii. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?

Solution:

This is the value of C, so we have that: $a+b+(A\bigcap B)+C=1$

$0.698+C=1$

$C=0.302$

30.2% probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons.

ANSWER: $30.2\%$