Question

A large number of adults are classified according to whether they were judged to need eyeglasses for reading and whether they actually used eyeglasses when reading. The proportions falling into the four categories are shown in the table. A single adult is selected from this group. Find the probabilities given here. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Used Eyeglasses } \\ \text { for Reading } \end{array} & \\ \hline \text { Judged to Need Eyeglasses } & \text { Yes } & \text { No } \\ \hline \text { Yes } & .44 & .14 \\ \text { No } & .02 & .40 \end{array} $$ a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not. d. An adult used glasses when they didn't need them.

Short answer

Question: Determine the following probabilities based on the given table of proportions: a) The adult is judged to need eyeglasses. b) The adult needs eyeglasses for reading but does not use them. c) The adult uses eyeglasses for reading whether he or she needs them or not. d) An adult used glasses when they didn't need them. Answer: a) P(Judged to need eyeglasses) = 0.58 b) P(Needs eyeglasses, does not use them) = 0.14 c) P(Uses eyeglasses for reading) = 0.46 d) P(Uses glasses, doesn't need them) = 0.02

Step-by-step solution

Scenario A: Probability the adult is judged to need eyeglasses

To find the probability that an adult is judged to need eyeglasses, we need to look at the table and sum the proportions of adults who were judged to need eyeglasses, regardless of whether they used them or not. This would include the adults in the "Yes" row of the "Judged to Need Eyeglasses" column. This probability can be calculated as: P(Judged to need eyeglasses) = P(Yes, Yes) + P(Yes, No) = 0.44 + 0.14

Scenario B: Probability the adult needs eyeglasses for reading but does not use them

To find the probability of this scenario, we just need to look at the table, specifically the cell where "Judged to Need Eyeglasses" is Yes, and "Used Eyeglasses for Reading" is No. This probability is directly given in the table: P(Needs eyeglasses, does not use them) = 0.14

Scenario C: Probability the adult uses eyeglasses for reading whether he or she needs them or not

For this scenario, we need to calculate the probability that an adult uses eyeglasses for reading. To do that, we will look at the table and sum the proportions of adults who used eyeglasses for reading, regardless of whether they were judged to need them or not. This would include the adults in the "Yes" column of the "Used Eyeglasses for Reading" row. This probability can be calculated as: P(Uses eyeglasses for reading) = P(Yes, Yes) + P(No, Yes) = 0.44 + 0.02

Scenario D: Probability an adult used glasses when they didn't need them

This scenario is about the probability that an adult used eyeglasses for reading when they were not judged to need them. We need to look at the table and find the cell where "Judged to Need Eyeglasses" is No, and "Used Eyeglasses for Reading" is Yes. This probability is directly given in the table as: P(Uses glasses, doesn't need them) = 0.02 Now we can compile our answers: a) P(Judged to need eyeglasses) = 0.44 + 0.14 = 0.58 b) P(Needs eyeglasses, does not use them) = 0.14 c) P(Uses eyeglasses for reading) = 0.44 + 0.02 = 0.46 d) P(Uses glasses, doesn't need them) = 0.02

Key concepts

Probability Calculation

When we talk about probability calculation, we're dealing with the likelihood that a certain event will occur. It's a fundamental aspect of probability theory that allows us to make predictions and informed decisions.

Consider a real-world example similar to the exercise regarding adults who use or do not use eyeglasses for reading. To calculate the probability that a randomly selected adult is judged to need eyeglasses, we sum the proportions who were judged to need eyeglasses. This process of summing, often referred to as total probability, is critical for understanding composite chances.

In the exercise, probability was calculated by summing two numbers given in a contingency table. This table is an organized representation of different outcomes and their associated probabilities. Understanding how to read and manipulate these tables is crucial in probability calculation. Doing this accurately ensures reliable predictions and insightful interpretations of events based on statistical data.

Statistical Data Interpretation

The field of statistical data interpretation involves making sense of numerical data and turning it into valuable information. It's like being a detective, where data gives you clues and interpretation lets you solve the case.

In the context of the exercise, interpreting the data correctly is paramount. For instance, interpreting the meaning behind the probability of an adult needing but not using eyeglasses (0.14 in our case) is more than just knowing the number; it involves understanding the implications of this behavior on vision health and preventive care.

Importance of Context

Context shapes the interpretation. In our example, knowing that 14% of adults need but don't use glasses can inform health campaigns or policies. It's not just a number—it reflects real-world tendencies and choices.

Visual Aids

Tools like graphs, charts, and tables enhance our ability to interpret statistical data. They provide a visual snapshot of the information, making it easier to analyze and understand patterns and relationships. Visual aids transform raw data into a picture that tells a story.

Probability Theory

At its core, probability theory is the mathematical framework that deals with the uncertainty of events. It's the foundation upon which all probability calculations and interpretations are built.

The theory provides us with the laws and principles needed to analyze random phenomena and underpins many disciplines from finance to physics.

Basic Concepts

Fundamental to probability theory are concepts like random variables, expected values, and variance. A random variable, for example, represents potential outcomes of a random process—like those found in the eyeglasses exercise. Expected values then allow us to calculate the 'center' or 'average' outcome we anticipate from a probability distribution.

Real-World Applications

More than just academic, these principles have tangible impacts. Deciding whether or not to carry an umbrella based on a weather forecast is a practical application of probability theory. Each aspect of the theory, from simple probabilities to complex models, gives us a lens through which we can predict and manage the uncertainty of the world around us.