Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The results of a recent television survey of American TV households revealed that 87 out of every 100 TV households have at least one remote control. What is the probability that a randomly selected TV household does not have at least one remote control?

Step-by-Step Solution

Answer

The probability that a randomly selected TV household does not have at least one remote control is \(\frac{13}{100}\) or 13%.

Step 1: Understand the given information

We are given that 87 out of every 100 TV households have at least one remote control. This can be represented as a proportion by dividing the number of households with at least one remote control (87) by the total number of households (100).

Step 2: Calculate the proportion of households with at least one remote control

The proportion of households with at least one remote control can be calculated as: \[\frac{87}{100}\]

Step 3: Find the complement of the proportion

The complement of a proportion is the remaining part that makes it equal to 1. In other words, it's the probability of the opposite event occurring. In this case, we want to find the probability of a TV household not having at least one remote control. To find the complement, subtract the proportion with at least one remote control from 1: \[1 - \frac{87}{100}\]

Step 4: Calculate the probability of not having at least one remote control

Now calculate the result: \[1 - \frac{87}{100} = \frac{13}{100}\] So the probability that a randomly selected TV household does not have at least one remote control is \(\frac{13}{100}\) or 13%.

Key Concepts

Complementary Events

In probability, complementary events are two outcomes that cover all possible scenarios. They are like two sides of a coin: if either one happens, the other cannot. For instance, when a TV household either has a remote control or doesn't, these two outcomes are complementary events.

Understanding complementary events is crucial for calculating probabilities because they add up to 1, or 100%. If the probability of one event is known, the complement, or opposite event, is simply 1 minus that probability. This concept is used in the step-by-step solution when calculating the probability of not having at least one remote control. If the probability of having one is 87 out of 100, then the probability of not having one is the complement: 13 out of 100.

Proportions

Proportions represent a part of the whole and are often expressed as fractions or percentages. They are useful for showing the size of one group relative to a total group. In the original exercise, calculating the proportion of TV households with and without a remote control helps in understanding the survey results.

To find proportions, divide the number of desired outcomes by the total number of outcomes. For example, the proportion of households with at least one remote control is found by dividing 87 by 100, creating the fraction \( \frac{87}{100} \) or 87%.

• This proportion indicates that 87% of TV households have a remote control.
• The remaining 13% represent those without, as shown through complementary events.

Proportions offer a clear way to compare parts of the data to the whole and develop insights from statistical surveys.

Statistical Survey

A statistical survey is a method for collecting and analyzing data from a group of individuals or households to understand patterns and trends. Surveys often inquire about preferences, behaviors, or opinions.

In the original exercise, the survey targets American TV households regarding ownership of remote controls. By polling 100 households, it estimates widespread behavior across the larger population. Surveys must be carefully designed to ensure accuracy, using representative samples and clear questions.

Statistical surveys help gather crucial data for decision-making in research, business, and policy. When results, as seen in the exercise, indicate a high proportion of TV households having remote controls, companies can tailor strategies to accommodate this trend or address the households without remote controls.

Applied Mathematics

Applied Mathematics uses mathematical methods and models to solve practical problems. It often involves applying concepts like probability, algebra, and statistical analysis to real-world scenarios.

In the context of the original exercise, applied mathematics provides the tools to interpret survey data and understand behaviors within populations. This involves estimating probabilities with proportions, enabling predictions and strategic planning.

In real life, businesses utilize these mathematical approaches to analyze market trends, optimize production, and enhance product distribution. By understanding the probability of remote control ownership, companies can make data-driven decisions. For example, they could decide whether to market universal remotes based on the proportion of households without an existing remote.