Problem 5 - 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 $$\text{is a widowed male.}$$

Step-by-Step Solution

Answer

The probability that a randomly selected individual is a widowed male is \(\frac{3}{242}\) or approximately 0.01 when expressed as a decimal to the nearest hundredth.

Step 1: Extract Required Data from the Table

From the data, it can be seen that the number of widowed males is 3 million. The total population exists in the bottom right section of the table as 242 million.

Step 2: Calculate the Probability

Probability is calculated as the number of desired outcomes (widowed males) divided by the total number of outcomes (total population). Therefore, the probability \(\frac{3}{242}\) represents the likelihood that a person randomly selected from the population is a widowed male.

Step 3: Express the Probability as a Decimal

To convert the fraction to a decimal, divide the numerator (3) by the denominator (242). Round the resulting decimal to the nearest hundredth if necessary.

Key Concepts

Marital Status

In statistics, **marital status** is a way to categorize the U.S. population based on whether individuals are married, never married, divorced, widowed, and so forth. In the given table, marital status classifications are essential for examining various population subsets, particularly when analyzing trends or behaviors associated with each group.
For example, when we analyze the probability of someone being a widowed male, we only consider the subset of the population marked as 'widowed' under the 'male' category.
Marital status data helps provide insights into social and economic trends, such as changes in marriage and divorce rates over time.

Fraction Simplification

When dealing with probabilities, simplifying fractions is a vital part of expressing the likeliness of any given event in the most simplified manner possible.
In the exercise, after determining that 3 out of the total 242 individuals are widowed males, we express this probability as a fraction:

• Fraction form: \( \frac{3}{242} \). It represents our desired event over the total possible events.

Though in this case \( \frac{3}{242} \) is already in its simplest form as both the numerator and the denominator don't share any common divisors other than 1, always check for simplification to ensure clarity and accuracy.

Decimal Conversion

Decimal conversion is the process of changing a fraction into a decimal format, which can sometimes help in understanding probabilities more intuitively.
To convert the fraction \( \frac{3}{242} \) into a decimal:

• Divide the numerator by the denominator: \( 3 \div 242 \approx 0.0123967 \).
• Round the result to the nearest hundredth, which gives you \( 0.01 \).

By converting the probability to a decimal, especially when it is rounded to a practical number of decimal places, it becomes easier to grasp, particularly if you are comparing it to other probabilities.

Population Statistics

Population statistics involves the analysis of data collected about the number of people, typically classified into subgroups such as gender and marital status.
This statistical analysis helps us to understand broader societal patterns and behaviors.
Such classifications help answer specific questions like "What is the probability of selecting a widowed male from the U.S. population in 2010?" which was determined using data from a sample of 242 million people. This exercise involves applying probability principles to real-world data, offering valuable insights into demographic trends and assisting in policy-making and economic planning. Population data thus remains a crucial resource for researchers and policymakers alike.