Question

A large cable TV company reports the following: \- \(80 \%\) of its customers subscribe to its cable TV service \- \(42 \%\) of its customers subscribe to its Internet service \- \(32 \%\) of its customers subscribe to its telephone service \(25 \%\) of its customers subscribe to both its cable TV and Internet service \(21 \%\) of its customers subscribe to both its cable TV and phone service \- \(23 \%\) of its customers subscribe to both its Internet and phone service \- \(15 \%\) of its customers subscribe to all three services Consider the chance experiment that consists of selecting one of the cable company customers at random. Find and interpret the following probabilities: a. \(P(\) cable TV only \()\) b. \(P(\) Internet \(\mid\) cable \(\mathrm{TV})\) c. \(P\) (exactly two services) d. \(P\) (Internet and cable TV only)

Short answer

The probabilities for respective cases are: a. \(P(\) Cable TV only \()\) = \(80 \% -25 \% -21 \% +15 \% = 49 \%); b. \(P(\) Internet \(\mid\) Cable TV \()\) = \(25 \% / 80 \% = 31.25 \%); c. \(P\) (exactly two services) = \(25 \% + 21 \% + 23 \% - 3*15 \% = 19 \%); d. \(P\) (Internet and Cable TV only) = \(25 \% - 15 \% = 10 \%.

Step-by-step solution

Represent the data

It helps to tabulate and represent the problem in a Venn Diagram for easier understanding. Remember that the percentages given have to be correctly placed in the right regions of the Venn Diagram.

Calculate P(Cable TV only)

To find probability of Cable TV only subscription, subtract the percentages of customers subscribing to cable TV along with other services from total Cable TV subscription percentage. Mathematically, this is given by \(P(\) Cable TV only \()\) = Total Cable TV - (Cable TV and Internet) - (Cable TV and Phone) + (all three services).

Calculate P(Internet|Cable TV)

Now, to find this probability, this is a case of conditional probability. Here, the total must be restricted to the Cable TV subscriptions because the condition is given as 'given Cable TV'. Therefore, \(P(\) Internet \(\mid\) Cable TV \()\) = (Cable TV and Internet) / Total Cable TV.

Calculate P(exactly two services)

For the probability of exactly two services, sum the percentages of consumers who subscribe exactly to two services. Remember that this excludes those who subscribe to all three services, hence subtract those taking all 3 services from the total.

Calculate P(Internet and Cable TV only)

In this case, subtract the customers who are subscribing to all three services from those who are subscribing to Internet and Cable TV. So, \(P(\) Internet and Cable TV only \()\) = (Cable TV and Internet) - (all three services).