Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

According to a survey conducted in 2004 of 1000 American adults with Internet access, one in four households plans to switch ISPs in the next 6 months. Of those who plan to switch, \(1 \%\) of the households are likely to switch to a satellite connection, \(27 \%\) to digital subscriber line (DSL), \(28 \%\) to cable modem, \(35 \%\) to dial-up modem, and \(9 \%\) don't know what kind of service provider they will switch to. a. What is the probability that a randomly selected survey participant who was planning to switch ISPs will switch to a dial-up modem connection? b. What is the probability that a randomly selected survey participant will upgrade to high-speed service (satellite, DSL, or cable)?

Step-by-Step Solution

Answer

a. The probability that a randomly selected survey participant who is planning to switch ISPs will choose a dial-up modem connection is \(P(\text{dial-up modem connection}) = 0.35\). b. The probability that a randomly selected survey participant will upgrade to high-speed service (satellite, DSL, or cable) is \(P(\text{high-speed service}) = 0.56\).

Step 1: Recall the probability of switching to a dial-up connection

From the given information, we know that \(35 \%\) of the households who intend to switch ISPs will choose a dial-up modem connection. This is the probability we are looking for.

Step 2: Write the probability as a fraction or decimal

Expressing this probability as a fraction or decimal, we get the answer as: \(P(\text{dial-up modem connection}) = \frac{35}{100} = 0.35\). b. Probability of upgrading to high-speed service (satellite, DSL, or cable)

Step 3: Find the probabilities of each high-speed service

First, let's find the probabilities of upgrading to each high-speed service (satellite, DSL, and cable) separately. They are given as follows: - Satellite connection: \(1 \%\) - DSL connection: \(27 \%\) - Cable modem connection: \(28 \%\)

Step 4: Calculate the combined probability

Since the households can upgrade to only one type of high-speed service, the events are mutually exclusive. Therefore, to find the probability that a household will upgrade to any of these three services, we can simply add their individual probabilities: \(P(\text{high-speed service}) = P(\text{satellite}) + P(\text{DSL}) + P(\text{cable})\)

Step 5: Substitute the given probabilities and calculate the answer

Substituting the given probabilities, we get: \(P(\text{high-speed service}) = \frac{1}{100} + \frac{27}{100} + \frac{28}{100} = \frac{56}{100} = 0.56\) So, the probability that a randomly selected survey participant will upgrade to a high-speed service (satellite, DSL, or cable) is \(0.56\).

Key Concepts

Survey Data Analysis

Survey data analysis is the process of collecting, examining, and interpreting data gathered from surveys. It helps to draw conclusions and make informed decisions. In this particular exercise, survey data collected from 1000 American adults reveals the preferences for various Internet Service Providers (ISPs).
When analyzing survey data:

• Identify the total number of respondents: Knowing this helps to calculate percentages and probabilities.
• Understand the distribution of responses: Examine each response category to ensure data accuracy.
• Find patterns in the data: Patterns can suggest trends, like which ISP types are preferred.

In our exercise, a key finding is that 25% of survey respondents intend to change ISPs. Understanding these trends can provide valuable insights into how consumer behavior may shift over time.

Internet Service Providers

Internet Service Providers, or ISPs, play a crucial role in connecting users to the internet. They offer different types of internet access such as satellite, DSL (Digital Subscriber Line), cable modem, and dial-up. Each type of service comes with its unique features and advantages:

• Satellite: Offers internet access through satellites but can be affected by weather conditions.
• DSL: A broadband connection using phone lines, offering a stable connection unaffected by neighbours' usage.
• Cable Modem: Uses cable TV lines for internet access and typically provides fast speeds.
• Dial-Up Modem: The oldest form of connection, slower and often used in areas without alternatives.

In the given survey, participants expressed their preferences regarding which ISP they plan to switch to. This information is valuable for ISPs to tailor their services to meet consumer needs.

Mutually Exclusive Events

Mutually exclusive events are scenarios where the occurrence of one event prevents the happening of another. In other words, both events cannot occur simultaneously. This concept is crucial in probability calculations, especially when dealing with survey outcomes.
In this exercise, the decision to switch to one type of high-speed service is mutually exclusive with choosing another. If a household selects DSL, it cannot simultaneously choose cable or satellite.

• For mutually exclusive events, you calculate the combined probability by adding the individual probabilities together.
• Formula: If A and B are mutually exclusive, then \(P(A \text{ or } B) = P(A) + P(B)\).

By adding the probabilities of switching to satellite, DSL, or cable, we determine the overall likelihood that a survey participant will choose to upgrade to a high-speed service. This illustrates the practical application of mutually exclusive events in probability theory.