Problem 1 - 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 is divorced.

Step-by-Step Solution

Answer

The probability that a randomly selected person is 'divorced' is \( \frac{12}{121} \) or 0.099 (rounded to three decimal places).

Step 1: Identify the Required Data from the Table

From the table, one can see that the total number of 'divorced' persons is 24 (from both genders combined). Also, the total population size, or the total number of people listed in the table, is 242.

Step 2: Compute the Probability

The probability of an event is given by the ratio of the favorable outcomes to the total outcomes. In this case, the favorable outcomes are the individuals who are 'divorced' and the total outcomes are the total number of people. Hence, the required probability is \( \frac{24}{242} \).

Step 3: Simplify the Resulting Fraction

The fraction \( \frac{24}{242} \) simplifies to \( \frac{12}{121} \). This is done by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Step 4: Convert the Simplified Fraction to a Decimal

By dividing 12 by 121, the approximate decimal representation of the fraction is obtained, which is 0.099 (rounded to three decimal places).

Key Concepts

Understanding the Probability of an Event

Probability is a fundamental concept in statistics that tells us how likely an event is to occur. When you hear the term "probability of an event," it refers to the measure of the chance that a particular outcome will happen. To find the probability, you divide the number of ways the event can occur by the total number of possible outcomes.
This gives you a ratio that can be expressed in different forms, such as fractions, decimals, or percentages. In this example, to find the probability that a randomly selected person is divorced, we look at the event where the person chosen is from the ‘divorced’ group.

Creating a Simplified Fraction

A simplified fraction makes numbers easier to work with by reducing them to their smallest form. To simplify a fraction, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
In our example, the fraction for the probability that a person is divorced started as \( \frac{24}{242} \). By dividing both the numerator and denominator by their GCD, which is 2, we reduced the fraction to \( \frac{12}{121} \).
It is always helpful to simplify fractions for ease of interpretation and computation.

Interpreting Decimal Representation

Decimal representation offers another format to express probability, making it often easier to understand at a glance. After simplifying the fraction, you can convert it to a decimal by performing the division \( 12 \div 121 \).
This yields approximately 0.099. Converting probabilities into decimals is common because it provides a straightforward way to compare likelihoods. Rounding to the nearest hundredth, which in this case is 0.10, simplifies communication, but keep track of what rounding means for accuracy in interpretation.

Counting Favorable Outcomes

Favorable outcomes are the specific outcomes of an event that meet the criteria of interest. Here, the favorable outcomes are the number of 'divorced' individuals identified in the table.
From the exercise, we learned that the number of people who are divorced equals 24, encompassing both males and females. These outcomes form the numerator in our probability fraction. Identifying favorable outcomes is a fundamental skill in problem-solving, providing a clear focus on what we are measuring.

Evaluating Total Outcomes

Total outcomes represent all possible results that can occur in a given scenario. It's essential to enumerate these accurately for precise probability calculations.
In this context, the total outcomes refer to the entire population size listed in the table, which is the denominator of our probability fraction. The total count amounts to 242 when considering all marital statuses across genders. Understanding the total outcomes ensures that probabilities are computed correctly and are representative of the entire dataset.