Problem 2 - Precalculus

Question

$$\begin{array}{|l|c|c|c|c|} \hline & \text { Married } & \text { Never } & \text { Divorced } & \text { Widowed } & \text { Total } \\ \hline \text { Male } & 65 & 40 & 10 & 3 & 118 \\ \hline \text { Female } & 65 & 34 & 14 & 11 & 124 \\ \hline \text { Total } & 130 & 74 & 24 & 14 & 242 \\ \hline \end{array}$$ 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 has never been married.

Step-by-Step Solution

Answer

The probability that the person has never been married is approximately 0.31 or 31% when expressed as a decimal, and \(74/242\) when expressed as a fraction.

Step 1: Understand the presented table

The table shows the breakdown of the population by marital status and gender. The total number of people, 242, is at the bottom right corner. The number of people who have never been married, 74, is mentioned under the 'Never' column in the 'Total' row. This column signifies those individuals who have never been married.

Step 2: Apply the formula for probability

The probability of an event A is given by the formula: Probability(A) = Number of desirable outcomes / Total number of outcomes. Here, the desirable outcome is 'the number of people who have never been married' and the total number of outcomes is the total population.

Step 3: Compute the probability

Insert the values: Probability = Number of people who have never been married / Total number of people = 74 / 242. Simplify this fraction.

Step 4: Convert to decimal

To convert the probability from a fraction to a decimal, divide the numerator by the denominator.

Key Concepts

Marital Status Statistics

Probability in statistics often deals with the likelihood of a particular event happening within a group or population. In this exercise, we're exploring the population's marital status, specifically focusing on those who have "never been married."
Understanding marital status statistics is essential when dealing with sociological or demographic data. In this table, data is separated by genders, male and female, as well as various marital statuses such as married, never married, divorced, and widowed.
Each category not only tells us how society is structured but also helps us calculate probabilities. For example:

• The total population is stated at 242.
• The number of individuals who have never been married stands at 74.

These statistics are fundamental in calculating the probability that a randomly selected person from this group falls under the 'never married' category, which we will simplify into fraction and decimal forms for easier comprehension and application in probability-related problems.

Fraction Simplification

Fractions are a foundational element in mathematics, representing parts of a whole. In statistics and probability, simplifying fractions makes them easier to comprehend and use in calculations.
To simplify a fraction, you must divide the numerator and denominator by their greatest common divisor (GCD). In our exercise, the probability of a person never being married is expressed as 74/242.
We need to simplify this:

• First, find the GCD of 74 and 242.
• The GCD of 74 and 242 is 2.
• Now, divide both the numerator and the denominator by 2.

When you divide, you get 37/121. Now, 37/121 is the simplest form of this probability.
Using simplified fractions for calculations ensures that your answers are both accurate and easy to understand, which is especially helpful when working with statistics. It also aids in the clear communication of results.

Converting Fractions to Decimals

Once you have a simplified fraction, converting it into a decimal can sometimes make results easier to interpret, especially when comparing probabilities.
Converting fractions to decimals involves division where the numerator is divided by the denominator.
In our scenario, the simplified probability fraction is 37/121. Divide 37 by 121:

• 37 ÷ 121 = 0.305785124.
• Round it to the nearest hundredth.

After rounding, we get approximately 0.31. This decimal representation allows for straightforward comparison with other probabilities and a better understanding of the statistics involved.
Decimals make probability analysis in real-world applications more tangible and less abstract, hence making statistics more accessible to everyone.