Question

A survey of magazine subscribers showed that 45.2% rented a car during the past 12 months for business reasons, 56% rented a car during the past 12 months for personal reasons, and 32% rented a car during the past 12 months for both business and personal reasons. (a) What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons? (b) What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal rea

Answer 1

Answer:

a. 0.692 or 69.2%; b. 0.308 or 30.8%.

Step-by-step explanation:

This is the case of the probability of the sum of two events, which is defined by the formula:

$\\ P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (1)

Where $\\ P(A \cup B)$ represents the probability of the union of both events, that is, the probability of event A plus the probability of event B.

On the other hand, $\\ P(A \cap B)$ represents the probability that both events happen at once or the probability of event A times the probability of event B (if both events are independent).

Notice the negative symbol for the last probability. The reason behind it is that we have to subtract those common results from event A and event B to avoid count them twice when calculating $\\ P(A \cup B)$.

We have to remember that a sample space (sometimes denoted as S) is the set of the all possible results for a random experiment.

Calculation of the probabilities

From the question, we have two events:

Event A: event subscribers rented a car during the past 12 months for business reasons.

Event B: event subscribers rented a car during the past 12 months for personal reasons.

$\\ P(A) = 45.2\%\;or\;0.452$

$\\ P(B) = 56\%\;or\;0.56$

$\\ P(A \cap B) = 32\%\;or\;0.32$

With all this information, we can proceed as follows in the next lines.

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

We have to use here the formula (1) because of the sum of two probabilities, one for event A and the other for event B.

Then

$\\ P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\\ P(A \cup B) = 0.452 + 0.56 - 0.32$

$\\ P(A \cup B) = 0.692\;or\;69.2\%$

Thus, the probability that a subscriber rented a car during the past 12 months for business or personal reasons is 0.692 or 69.2%.

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

As we can notice, this is the probability for the complement event that a subscriber did not rent a car during the past 12 months, that is, the probability of the events that remain in the sample space. In this way, the sum of the probability for the event that a subscriber rented a car plus the event that a subscriber did not rent a car equals 1, or mathematically:

$\\ P(\overline{A \cup B}) + P(A \cup B)= 1$

$\\ P(\overline{A \cup B}) = 1 - P(A \cup B)$

$\\ P(\overline{A \cup B}) = 1 - 0.692$

$\\ P(\overline{A \cup B}) = 0.308\;or\;30.8\%$

As a result, the requested probability for a subscriber that did not rent a car during the past 12 months for either business or personal reasons is 0.308 or 30.8%.

We can also find the same result if we determine the complement for each probability in formula (1), or:

$\\ P(\overline{A}) = 1 - P(A) = 1 - 0.452 = 0.548$

$\\ P(\overline{B}) = 1 - P(B) = 1 - 0.56 = 0.44$

$\\ P(\overline{A \cup B}) = 1 - P(A \cup B) = 1 - 0.32 = 0.68$

Then

$\\ P(\overline{A \cup B}) = P(\overline{A}) + P(\overline{B}) - P(\overline{A\cap B})$

$\\ P(\overline{A \cup B}) = 0.548 + 0.44 - 0.68$

$\\ P(\overline{A \cup B}) = 0.308$