A 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. What is the 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?

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 reasonsThis 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 CWe 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.