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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)

Step by step solution

01

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.

02

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).

03

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.

04

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.

05

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability

Conditional Probability is a concept used to determine the likelihood of an event occurring given that another event has already occurred. In simpler terms, it limits our sample space to only those outcomes linked to a prior event. It's like asking: out of all the customers who subscribe to Cable TV, how many also use the Internet?
This scenario uses a formula:

• \( P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \)

where \( A \) is the Internet subscription and \( B \) is the Cable TV subscription.
Using the given data, to find the probability of Internet users among those who are Cable TV subscribers, simply divide the percentage of customers using both services by the total percentage of Cable TV subscribers. This way, you can narrow down your focus to a specific group of customers.

Venn Diagram

A Venn Diagram is a helpful visual tool used for illustrating relationships between different sets or groups. It consists of circles that overlap to represent groups sharing common elements.
In the context of the TV company problem, the Venn Diagram helps visualize which customers subscribe to which services. By correctly placing the given percentage values, one can easily identify:

• The overlapping regions where customers subscribe to multiple services.
• Regions exclusive to one service to highlight unique subscriptions.

Visual aids like Venn Diagrams make it simpler to initially sort complimentary data, and then one can proceed to solve probability problems more systematically and logically.

Step-by-Step Solution

Breaking problems down into smaller chunks makes them way more approachable. This is what a Step-by-Step Solution does, guiding you through each part of the problem systematically. Here’s how each part contributes:

• Step 1: Start by representing the given data, which sets the groundwork. Understanding the layout ensures that calculations are built on a clear base.
• Step 2: Calculating \( P(\text{Cable TV only}) \) involves subtracting overlaps, in this case, other service subscriptions. Lay out your calculation in a straightforward manner: Total CTV minus those within other shared service brackets plus all three services overlapping due to over-subtraction initially.
• Step 3: Finding \( P(\text{Internet | Cable TV}) \) restricts our whole sample universe to only Cable TV subscribers, nicely displaying Conditional Probability.
• Step 4: Calculating for exactly two services involves understanding overlap and eliminating triples. The principle - reduce the number of subscribers in all three from those in any pairs.
• Step 5: Completing \( P(\text{Internet and Cable TV only})\) mirrors step four but with a focus on two specific services only, incorporating subtraction of triple overlaps.

This guided breakdown makes complex multivariate problems much easier to analyze and solve.

Data Representation

Accurate Data Representation is foundational for probability problems like this one. It involves visually or numerically representing the given information clearly and precisely.
Here’s why this is vital:

• Graphical representations, such as Venn Diagrams, lay out intersection and union sets visually, simplifying the comprehension of group relationships and overlap.
• Numerical representation, on the other hand, gives a clear view of all values involved. This helps cross-check long or complex calculations.

Being able to use and understand these representations is crucial for solving multi-event probability problems thoroughly and accurately. With clear data representation, you can pivot into calculations with a crystal-clear picture of relationships, boosting the accuracy and efficiency of your problem-solving approach.