91Ó°ÊÓ

A large cable company reports the following: \(80 \%\) of its customers subscribe to cable TV service \(42 \%\) of its customers subscribe to Internet service \(32 \%\) of its customers subscribe to telephone service \(25 \%\) of its customers subscribe to both cable TV and Internet service \(21 \%\) of its customers subscribe to both cable TV and phone service \(23 \%\) of its customers subscribe to both 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. In Exercise \(5.53,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. \(P(\) cable TV only) b. \(P\) (Internet \(\mid\) cable TV) c. \(P(\) exactly two services \()\) d. \(P\) (Internet and cable TV only)

Step by step solution

01

Part A: Probability of Cable TV Only

To find the probability of a customer subscribing to Cable TV only, we will use the following formula: \[P(\text{Cable TV only}) = P(\text{Cable TV}) - P(\text{Cable TV and Internet}) - P(\text{Cable TV and Phone}) + P(\text{All Three})\] Now, let's substitute the given values: \(P(\text{Cable TV only}) = \frac{80}{100} - \frac{25}{100} - \frac{21}{100} + \frac{15}{100}\) \(P(\text{Cable TV only}) = \frac{49}{100}\) So, the probability of a customer subscribing to Cable TV only is 0.49.

02

Part B: Conditional Probability of Internet given Cable TV

To find the conditional probability of a customer subscribing to Internet given they have Cable TV, we will use the following formula: \[P(\text{Internet | Cable TV}) = \frac{P(\text{Cable TV and Internet})}{P(\text{Cable TV})}\] Now, let's substitute the given values: \(P(\text{Internet | Cable TV}) = \frac{\frac{25}{100}}{\frac{80}{100}}\) \(P(\text{Internet | Cable TV}) = \frac{25}{80}\) So, the conditional probability of a customer subscribing to Internet given they have Cable TV is \(\frac{25}{80}\) or 0.3125.

03

Part C: Probability of Exactly Two Services

To find the probability of a customer subscribing to exactly two services, we will add the probabilities of each pair combination without including all three services at once: \[P(\text{Exactly Two Services}) = P(\text{Cable TV and Internet only}) + P(\text{Cable TV and Phone only}) + P(\text{Internet and Phone only})\] Now, let's subtract the All Three probability from each combination: \(P(\text{Cable TV and Internet only}) = \frac{25}{100} - \frac{15}{100}\) \(P(\text{Cable TV and Phone only}) = \frac{21}{100} - \frac{15}{100}\) \(P(\text{Internet and Phone only}) = \frac{23}{100} - \frac{15}{100}\) \[P(\text{Exactly Two Services}) = \frac{10}{100} + \frac{6}{100} + \frac{8}{100}\] \(P(\text{Exactly Two Services}) = \frac{24}{100}\) So, the probability of a customer subscribing to exactly two services is 0.24.

04

Part D: Probability of Internet and Cable TV Only

To find the probability of a customer subscribing to both Internet and cable TV only, we need to subtract the probability of all three services from the probability of cable TV and Internet: \[P(\text{Internet and Cable TV only}) = P(\text{Cable TV and Internet}) - P(\text{All Three})\] Now, let's substitute the given values: \(P(\text{Internet and Cable TV only}) = \frac{25}{100} - \frac{15}{100}\) \(P(\text{Internet and Cable TV only}) = \frac{10}{100}\) So, the probability of a customer subscribing to both Internet and Cable TV only is 0.1.

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Key Concepts

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

Conditional Probability

Conditional probability helps us find the probability of an event occurring given that another event has already occurred. It is a crucial concept when dealing with interconnected events. For instance, when trying to determine the likelihood of a customer subscribing to the Internet service given they already subscribe to Cable TV. The formula for conditional probability is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
In this situation, \(P(A|B)\) represents the probability of event A occurring given that event B has occurred. Here, A is subscribing to the Internet and B is having Cable TV. By using this formula, one can determine how existing subscriptions (like Cable TV) affect the likelihood of other subscriptions (such as Internet).
This concept is essential in understanding the interconnectedness of events, especially in customer behavior analysis.

Probability Formulas

Probability formulas allow us to mathematically compute the likelihood of various outcomes. These formulas can address complex probabilities such as overlapping and independent events. For this exercise, some basic probability formulas are essential:

• Addition Rule: This is used when finding the probability of either event A or event B occurring, but modifications are needed for overlapping events.
• Subtraction Rule: When events overlap, subtract the overlapping section to avoid double-counting.

The formula for finding the probability of a single service, like Cable TV only, takes into account overlapping services:\[ P(\text{Cable TV only}) = P(\text{Cable TV}) - P(\text{Cable TV and Internet}) - P(\text{Cable TV and Phone}) + P(\text{All Three}) \]This formula effectively isolates the probability of just one service by removing overlap issues.

Hypothetical Table

A hypothetical table is a powerful tool to visually organize probabilities, especially when dealing with multiple categories or overlapping groups. This table uses assumed numbers, like 1000 customers, to break down the given percentages into tangible figures, simplifying complex percentage data into manageable parts.

• Begin by converting percentages into actual counts based on the hypothetical total. For example, 80% of 1000 customers subscribing to Cable TV translates to 800 customers.
• Break down overlapping sections using given percentages and fill in the corresponding cells in the table.
• Use the table to visually separate and calculate probabilities for exclusive groups, combinations, and total possible outcomes.

This method simplifies complex probability calculations and makes it easier to see and solve problems involving multiple subscriptions. It is a highly intuitive approach often paired with theoretical formulas for clarity.