Question

A large cable company reports that \(42 \%\) of its customers subscribe to its Internet service, \(32 \%\) subscribe to its phone service, and \(51 \%\) subscribe to its Internet service or its phone service (or both). a. Use the given probability information to set up a "hypothetical \(1000 "\) table. b. Use the table to find the following: i. the probability that a randomly selected customer subscribes to both the Internet service and the phone service. ii. the probability that a randomly selected customer subscribes to exactly one of the two services.

Short answer

a. Assume there are 1000 customers. Based on the given proportions: 420 subscribe to the Internet service, 320 subscribe to the phone service, and 510 subscribe to either the Internet service or the phone service (or both). Then, it can be calculated that 230 customers subscribe to both services, and 280 customers subscribe to exactly one service. b. Therefore, the probability that a randomly selected customer subscribes to both the Internet service and the phone service is 0.23, and the probability that a randomly selected customer subscribes to exactly one of the two services is 0.28.

Step-by-step solution

Create a hypothetical 1000 table

Start by assuming there are 1000 customers (this makes working with percentages easier). Based on the given proportions: \n\n- 420 customers subscribe to the Internet service (42% of 1000), \n\n- 320 customers subscribe to the phone service (32% of 1000), and \n\n- 510 customers subscribe to either the Internet service or the phone service (or both) (51% of 1000).

Calculate the number of customers who subscribe to both services

Next, calculate the number of customers that subscribe to both services. This can be obtained by adding the number of customers who subscribe to the Internet and phone service, and then subtracting the number of customers who subscribe to either one. This gives \(420 (Internet service) + 320 (phone service) - 510 (either service) = 230 (both services). \) So, 230 out of 1000 customers subscribe to both services.

Calculate the number of customers who subscribe to exactly one service

You can calculate the number of customers who subscribe to exactly one service by subtracting the number of customers who subscribe to both services from the total number of customers who subscribe to either service. This gives \(510 (either service) - 230 (both services) = 280 (exactly one service)\). Therefore, 280 out of 1000 customers subscribe to exactly one service.

Convert the customer numbers into probabilities

Finally, you convert these numbers into probabilities. For the probability that a customer subscribes to both services, divide the number of customers who subscribe to both services by the total number of customers, which gives 230/1000 = 0.23. For the probability that a customer subscribes to exactly one service, divide the number of customers who subscribe to exactly one service by the total number of customers, which gives 280/1000 = 0.28.

Key concepts

Hypothetical 1000 Table

Understanding probability can sometimes be challenging, but a hypothetical 1000 table simplifies it. Imagine that you have a group of 1000 people or objects; when we discuss probabilities in percentages, each percentage point represents 10 members of this hypothetical group.

For example, if a company reports that 42% of its customers use its Internet service, we visualize this as 420 out of 1000 customers in our hypothetical scenario. Similarly, 32% for phone service becomes 320 out of 1000. Setting up such a table makes it straightforward to calculate combined probabilities and visualize overlapping groups.

When given the proportion of customers using either service, we face potential overlaps. The company reports 51%, or 510 customers using one service or both. Using a table, we can easily subtract the sum of individual services from this figure to find how many use both services. This method also proves invaluable in understanding more complex probability scenarios.

Joint Probability

Joint probability is a key concept when we look at how likely two events are to happen at the same time. Formally, it's the probability that two events, A and B, occur together. To calculate it, we take into account the individual probabilities of each event and any overlap between them.

In our example, the joint probability would represent customers who subscribe to both Internet and phone services. The calculation process involves finding the number of customers with both services — here, 230 out of 1000, or 23%. This value is our joint probability, indicating a significant overlap between the two services. It's also a great example to show how businesses can use joint probabilities to understand their customer base and plan bundles or promotions effectively.

Mutually Exclusive Events

While joint probability deals with events that can occur at the same time, mutually exclusive events cannot. Two events are mutually exclusive if the occurrence of one event means the other cannot occur simultaneously. A classic example would be tossing a coin; it cannot land on both heads and tails at the same time.

In our cable company scenario, if there were a group of customers who could only subscribe to either Internet or phone service, but not both, these would be mutually exclusive events. Knowing this is critical when analyzing probabilities, as it affects the calculations. For events that aren't mutually exclusive (like our Internet and phone services example), the total probability can exceed the probabilities of the individual events, reflecting the overlap. Clearly, the concept of mutually exclusive events is central to a proper understanding of probability and is a cornerstone in interpreting data correctly.