Problem 10 - Algebra and Trigonometry

Question

Shown again is the table indicating the marital status of the U.S. population in 2010. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises \(1-10 .\) Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth $$ \begin{array}{llllll} {} & {} & {\text { Never}} \\ {} & {\text {Married}} & {\text { Married }} & {\text { Divorced }} & {\text { Widowed }} & {\text { Total }} \\ \hline \text { Male } & {65} & {40} & {10} & {3} & {118} \\ \hline \text { Female } & {65} & {34} & {14} & {11} & {124} \\ \hline \text { Total } & {130} & {74} & {24} & {14} & {242} \end{array} $$ If one person is randomly selected from the population described in the table, find the probability, expressed as a simplified fraction and as a decimal to the nearest hundredth, that the person $$\text{Among women, find the probability of selecting a married person.}$$

Step-by-Step Solution

Answer

The probability of selecting a married woman from the population is approximately \( \frac{65}{124} \) or 0.52 to the nearest hundredth.

Step 1: Identify relevant data

From the table, it's clear that among women, there are 65 million married. The total number of women is 124 million.

Step 2: Calculate Probability

The probability of an event occurring is calculated by dividing the number of ways that event can occur (in this case, a woman being married) by the total number of possible outcomes (in this case, the total number of women). This can be represented as \( P(E) = \dfrac{n(E)}{n(S)} \), where \( P(E) \) is the probability of the event E (a woman being married), \( n(E) \) is the number of outcomes where the event E occurs (a woman being married), and \( n(S) \) is the total number of outcomes (the total number of women). In this case, the calculation for the probability would be \( P(\text{Married}) = \dfrac{65}{124} \).

Step 3: Simplify the Fraction and Convert to Decimal

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor which is 1, so the simplified fraction remains \( \dfrac{65}{124} \). And to express this probability as a decimal, we divide the numerator (65) by the denominator (124). The answer to two decimal places would be 0.52

Key Concepts

Marital Status Statistics

Understanding marital status statistics involves looking at data sets that show how many individuals fall into categories such as married, never married, divorced, and widowed. This type of data is crucial for understanding social and demographic patterns. In this exercise, we have a table showing the marital status of the U.S. population in 2010, categorized by gender. Analyzing such data helps identify trends and shifts in family structures over time, which can inform policy and social services. By studying these statistics, students can understand patterns in the population, compare the numbers of different marital statuses, and draw conclusions about societal behaviors and norms.

Fractions and Decimals

Fractions and decimals are two different ways to represent numbers and probabilities. A fraction consists of a numerator and a denominator, showing a part of a whole. Conversely, a decimal expresses parts of a whole using a base-ten system.In the context of probability, fractions are often preferred because they provide an exact number. However, decimals are also useful, particularly when we want an approximate value that's easy to interpret and compare.For example, in our probability calculation, we initially express the likelihood of an event as a fraction \( \frac{65}{124} \). To make this probability more relatable, we convert it into a decimal, 0.52. This helps in understanding and communicating statistical information effectively.

Simplified Fractions

Simplifying fractions is the process of reducing them to their smallest form without changing their value. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).While the given fraction \( \frac{65}{124} \) is already simplified because there is no number other than 1 that divides both 65 and 124, learning how to simplify fractions is an essential skill in math. It makes calculations easier and results clearer.Simplification helps in expressing probabilities in their most elegant form, which is crucial for better understanding and communication in mathematics.

Probability Calculation

Probability calculation is a fundamental concept in statistics, representing the chance of an event occurring. It's expressed as a ratio of the number of favorable outcomes to the total number of possible outcomes.In our exercise, to find the probability of selecting a married woman from the female population, we use the fraction \( \frac{65}{124} \). This is calculated by taking the number of married women, 65, and dividing it by the total number of women, 124.

• Importance: Probability helps predict outcomes in uncertain situations.
• Simplification: We express the result as a simplified fraction and then convert it to a decimal for easier interpretation.
• Usability: Understanding probability calculations assists in making informed decisions.

By exploring these calculations, students grasp not just how to find probabilities but why they are valuable for analyzing real-world scenarios.