Problem 38 - College Algebra with Modeling and Visualization

Question

\(\quad\) In \(2004,\) the U.S. death rate per \(100,000\) people was \(817 .\) What is the probability that a person selected at random died during \(2004 ?\) (Source: Depertment of Health and Human Services.)

Step-by-Step Solution

Answer

The probability is 0.00817, or 0.817%.

Step 1: Convert Death Rate to Probability

The given death rate is 817 deaths per 100,000 people. To convert this to a probability, interpret it as a fraction of the population. So, the probability is \( \frac{817}{100,000} \).

Step 2: Simplify the Fraction

Simplifying \( \frac{817}{100,000} \) involves dividing both the numerator and the denominator by 100000 to express this as a decimal value. This gives \( 0.00817 \).

Step 3: Interpret the Probability

The probability of a randomly selected person dying in 2004, based on the death rate, is \( 0.00817 \), or 0.817%. This means less than 1% chance.

Key Concepts

Death Rate

Understanding death rate is crucial for interpreting data on population health. The death rate is typically expressed as the number of deaths per a certain number of people in a population, often per 100,000 for clarity in large populations. In this exercise, the death rate is given as 817 per 100,000 people.
This means if you had 100,000 people, 817 would have died that year.
It's a standardized unit that allows for easier comparisons between different regions or populations.
Death rates are essential for various analyses, such as assessing the effectiveness of public health campaigns, identifying health disparities, and planning healthcare resources.
Understanding this data can illuminate broader patterns in a society’s health, guiding policy and resource allocation.
Keep in mind that while death rates provide a snapshot of mortality in a population, they do not reflect the cause of deaths or other demographic factors such as age and gender.

Fraction Simplification

Fraction simplification is a method used to reduce fractions to their simplest form. In this exercise, we have a fraction representing a probability: \( \frac{817}{100,000} \).
Simplifying fractions makes them easier to understand and work with.
To simplify \( \frac{817}{100,000} \), you can divide both the numerator and the denominator by their greatest common divisor. However, in this case, the fraction can be immediately converted to a simpler form by recognizing its components.
In simpler terms, for a practical application such as this, converting directly to a decimal is straightforward and useful.
Here are some key steps in simplifying fractions:

• Identify the greatest common factor of both numbers.
• Divide the numerator and the denominator by this factor.
• Once simplified, the fraction can be used in calculations or reported clearly.

Decimal Conversion

Decimal conversion involves changing a fraction into a number with a total value between whole numbers, useful when interpreting probabilities.
From the solution, we take the fraction \( \frac{817}{100,000} \) and convert it into the decimal 0.00817.
Converting fractions like these to decimals can make probability easier to understand at a glance because decimals provide a more intuitive sense of magnitude and comparison.
Here are some tips for converting fractions to decimals:

• Divide the numerator by the denominator using a calculator or long division.
• Pay attention to the place value, since each digit after the decimal point represents a fractional part (tenths, hundredths, etc.).
• Use decimal notation in probability to quickly assess small chances, especially when the number is less than 1.

In this example, the conversion shows that there is a 0.817% chance that a person selected at random died during 2004. This small probability is easily understood in everyday terms, reinforcing the death rate's impact.