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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. Find and interpret the following 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

Determine the probability of cable TV only

P(cable TV only) can be calculated as the total percentage of customers who subscribe to cable - the percentage who also subscribe to other services. \(P(\text{cable TV only}) = P(\text{cable TV}) - P(\text{cable and internet}) - P(\text{cable and phone}) + P(\text{all three})\)Plugging in the given values:\(P(\text{cable TV only}) = 0.80 - 0.25 - 0.21 + 0.15 = 0.49\)

02

Determine P(Internet | cable TV)

The probability of a customer subscribing to the internet given that they already subscribe to cable TV can be found using the definition of conditional probability. \(P(\text{Internet | cable TV}) = \frac{P(\text{cable TV and internet})}{P(\text{cable TV})}\)Substituting the given values:\(P(\text{Internet | cable TV}) = \frac{0.25}{0.80} = 0.3125\)

03

Determine Probability of exactly two services

The probability of a customer subscribing to exactly two services can be found as sum of the probabilities of subscribing to each pair of services minus the probability of subscribing to all three.\(P(\text{exactly two services}) = [P(\text{cable and internet}) + P(\text{cable and phone}) + P(\text{internet and phone})] - 2*P(\text{all three})\)Filling the given values:\(P(\text{exactly two services}) = [0.25 + 0.21 + 0.23] - 2*0.15 = 0.34\)

04

Determine Probability for Internet and cable TV only

This is the probability that a customer subscribes to Internet and cable TV but not telephone service. \(P(\text{Internet and cable TV only}) = P(\text{cable and internet}) - P(\text{all three})\)Substituting the given values:\(P(\text{Internet and cable TV only}) = 0.25 - 0.15 = 0.10\)

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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 understand the likelihood of an event occurring, given that another related event has already happened. This is often written as \(P(A \mid B)\), which means "the probability of A given B".In our exercise, we calculated the conditional probability of a customer subscribing to the internet service given they are already cable TV subscribers:

• The probability of both subscribing to cable and internet: \(P(\text{cable TV and internet}) = 0.25\)
• The probability of subscribing to cable TV: \(P(\text{cable TV}) = 0.80\)

Putting this into the formula for conditional probability gives:\[P(\text{Internet} \mid \text{cable TV}) = \frac{0.25}{0.80} = 0.3125\]This means there is a 31.25% chance that a cable TV subscriber also subscribes to the internet. Conditional probability is a powerful tool for understanding relationships between events.

Joint Probability

Joint probability helps us find the probability of two events occurring together at the same time. In mathematical terms, this is represented as \(P(A \cap B)\), the probability of both A and B happening. From the exercise, the joint probability concept is applied as:

• The probability of subscribing to both cable TV and internet: \(P(\text{cable TV and internet}) = 0.25\)
• The probability of subscribing to both internet and phone: \(P(\text{internet and phone}) = 0.23\)

These values were used to breakdown probabilities of different service combinations, such as calculating exactly two services. Joint probabilities are useful for determining overlaps and understanding combined likelihoods.

Probability Distribution

A probability distribution gives us a full view of all possible outcomes of a chance experiment and their corresponding probabilities. Often represented in tables or graphs, probability distributions help visualize the allocation of probabilities across events. In the cable company exercise, we derived a distribution showing how

• 80% subscribed to cable,
• 42% to internet, and
• 32% to phone services.

Furthermore, intersections like cable and internet subscribers (25%), allow us to comprehend how subscription overlaps. Recognizing these distributions assists in comprehensive analyses of combined and exclusive service subscriptions.

Probability Rules

Probability rules form the backbone of calculating probabilities accurately. These rules provide guidance on how to combine and evaluate different events:

• Addition Rule: Used when calculating the probability of either of two events. For two overlapping events, \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
• Multiplication Rule: For independent events, it helps find the probability of both occurring, \(P(A \cap B) = P(A) \cdot P(B)\).

In our specific calculations, these rules help differentiate between subscribing to 鈥渆xactly two鈥� services and being able to add or subtract probabilities appropriately, such as avoiding double-counting for subscribers of all three services. Understanding these core rules allows the seamless application of conditional and joint probabilities in complex scenarios.