Question

A large cable company reports that \(80 \%\) of its customers subscribe to its cable TV service, \(42 \%\) subscribe to its Internet service, and \(97 \%\) subscribe to at least one of these two services. (Hint: See Example 5.6\()\) a. Use the given probability information to set up a "hypothetical \(1000 "\) table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer subscribes to both cable TV and Internet service. ii. the probability that a randomly selected customer subscribes to exactly one of these services.

Short answer

a) The '1000 customer' table has 250 customers subscribing to both services, 550 subscribing to cable TV only and 170 subscribing to the Internet only. b) i) The probability that a customer subscribes to both services is 25% and ii) the probability that a customer subscribes to exactly one service is 72%.

Step-by-step solution

Construct the hypothetical 1000 table

Assume there are 1000 customers. The number that subscribe to the cable TV service is \(80 \% \times 1000 = 800\), the number that subscribe to the Internet service is \(42 \% \times 1000 = 420\), and the number that subscribe to either one of these services is \(97 \% \times 1000 = 970\). From this, it can be found that the number of customers who subscribe to both services is \(800 + 420 - 970 = 250\). Therefore, 250 customers subscribe to both, 550 subscribe to cable TV only and 170 subscribe to Internet only.

Find the probability for a customer subscribing to both services

The probability that a randomly selected customer subscribes to both services can be calculated by dividing the number of customers subscribing to both services by the total number of customers: \( \frac{250}{1000} = 0.25 \) or 25 \%.

Find the probability for a customer subscribing to exactly one service

The probability that a randomly selected customer subscribes to exactly one service can be calculated by adding the probabilities of subscribing to only one of the two services: \( \frac{550 (TV only) + 170 (Internet only)}{1000 (total)} = 0.72 \) or 72\%.