Question

1. A survey of local car dealers revealed that 64% of all cars sold last month had CD players,

28% had alarm systems, and 22% had both CD players and alarm systems.

a.
What is the probability one of these cars selected at random had neither a CD player nor
an alarm system?

b. What is the probability that a car had a CD player unprotected by an alarm system?


C.
What is the probability a car with an alarm system had a CD player?

D.
Are having a CD player and an alarm system disjoint events? Explain.

Answer 1

The probability of a car having neither a CD player nor an alarm system is 0.78. The probability of a car having a CD player without an alarm system is 0.42. The probability of a car with an alarm system having a CD player is 0.79.

Step-by-step explanation:

To answer these questions, we need to use the concept of probability and set theory. Let's go step by step:

a. The probability that a car selected at random had neither a CD player nor an alarm system is calculated by subtracting the probability of having both CD player and alarm system from 1. So, P(neither) = 1 - P(both) = 1 - 0.22 = 0.78.

b. The probability that a car had a CD player unprotected by an alarm system is calculated by subtracting the probability of having both CD player and alarm system from the probability of having a CD player. So, P(CD player unprotected) = P(CD player) - P(both) = 0.64 - 0.22 = 0.42.

c. The probability that a car with an alarm system had a CD player is calculated by dividing the probability of having both CD player and alarm system by the probability of having an alarm system. So, P(CD player | alarm system) = P(both) / P(alarm system) = 0.22 / 0.28 = 0.79.

d. No, having a CD player and an alarm system are not disjoint events because there are cars that have both CD player and alarm system. Disjoint events are events that cannot happen at the same time.

Answer 2

To answer these questions, we can use the following information from the survey:

64% of all cars sold last month had CD players
28% of all cars sold last month had alarm systems
22% of all cars sold last month had both CD players and alarm systems
a. To find the probability that one of these cars selected at random had neither a CD player nor an alarm system, we need to subtract the probability that it had either a CD player, an alarm system, or both from 1. The probability that a car has either a CD player, an alarm system, or both is the sum of the probabilities of each of these events, which is 64% + 28% - 22% = 70%. Therefore, the probability that a car has neither a CD player nor an alarm system is 1 - 70% = 30%.

b. To find the probability that a car had a CD player but no alarm system, we can subtract the probability that it had both a CD player and an alarm system from the probability that it had a CD player. The probability that a car had both a CD player and an alarm system is 22%, so the probability that a car had a CD player but no alarm system is 64% - 22% = 42%.

c. To find the probability that a car with an alarm system had a CD player, we can use the formula for conditional probability: P(CD player | alarm system) = P(CD player and alarm system) / P(alarm system). The probability that a car had both a CD player and an alarm system is 22%, and the probability that it had an alarm system is 28%, so the probability that a car with an alarm system had a CD player is 22% / 28% = 78.6%.