Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

According to Mediamark Research, 84 million out of 179 million adults in the United States correct their vision by using prescription eyeglasses, bifocals, or contact lenses. (Some respondents use more than one type.) What is the probability that an adult selected at random from the adult population uses corrective lenses?

Step-by-Step Solution

Answer

The probability that a randomly selected adult from the adult population in the United States uses corrective lenses is approximately 0.46927 or 46.93%.

Step 1: Identify the given information

We are given the following information: - Total number of adults in the United States: 179 million - Number of adults using corrective lenses: 84 million

Step 2: Calculate the probability

To find the probability that a randomly selected adult uses corrective lenses, we can use the formula: probability = \(\frac{\text{number of desired outcomes}}{\text{total number of possible outcomes}}\) In this case, the number of desired outcomes is the number of adults using corrective lenses (84 million), and the total number of possible outcomes is the total number of adults in the United States (179 million). probability = \(\frac{84 \,\text{million}}{179 \,\text{million}}\)

Step 3: Simplify the expression

Now, we can simplify the expression and get the probability in decimal form. probability = \( \frac{84}{179}\) probability ≈ 0.46927 The probability that a randomly selected adult from the adult population in the United States uses corrective lenses is approximately 0.46927 or 46.93%.

Key Concepts

Corrective Lenses Statistics

When discussing statistics related to corrective lenses in the United States, it's essential to understand the context and significance of these numbers. Corrective lenses include eyeglasses, bifocals, or contact lenses. It reflects a common need among the population to enhance vision clarity.

For instance, according to a study, out of 179 million adults in the U.S., 84 million are reported to use some form of corrective lenses. This amounts to a substantial portion of the population, indicating a widespread reliance on optical correction for basic visual functions.

These types of statistics are crucial for various stakeholders, including healthcare providers, policymakers, and the lens-manufacturing industry. Understanding these numbers helps in resource allocation and planning for public health strategies.

Mathematical Reasoning

In solving probability-related problems, a clear grasp of mathematical reasoning is vital. This exercise involves reasoning through the relationships between given values to determine a probability.

The first step in the process involved identifying the total number of adults and those using corrective lenses. This involves a clear description of a part (corrective lenses users) compared to a whole (total adult population).

Once identified, you are tasked with organizing this information in a manner that can be manipulated mathematically – here, using the probability formula. This formula, comparing the number of favorable outcomes to the total possible outcomes, is a foundational concept in probability politics. - Recognize relevant data: Start by identifying essential numbers and what they represent. - Establish relationships: See how these numbers interact to express broader realities in probability. - Apply mathematical concepts: Use established formulas to derive meaningful statistics.

Probability Calculation

Probability calculation is a fundamental aspect of statistics and is key in predicting outcomes in uncertain situations. To compute probability, one needs a clear understanding of the formula used: \[\text{Probability} = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \]

In our example, the 'favorable outcome' is an adult using corrective lenses, which is 84 million. The 'total possible outcomes' represents the entire adult population, which is 179 million.

By placing the values into our formula, we find:\[\text{Probability} = \frac{84}{179} \]This results in approximately 0.46927, meaning there's a 46.93% chance a randomly chosen adult uses corrective lenses. Such computations help us understand events' likelihood in statistical terms, critical for effective decision-making.

In sum, probability calculation is a powerful tool to anticipate and understand data trends and behaviors within a population. Understanding this helps in numerous fields, including health, marketing, and engineering.