Problem 13 - College Algebra

Question

A die is rolled. Find the probability of getting an odd number.

Step-by-Step Solution

Answer

The probability of getting an odd number when rolling a die is \( \frac{1}{2} \) or 50%.

Step 1: Identify total and desired outcomes

A six-faced die has six outcomes: 1, 2, 3, 4, 5, 6. The desired outcomes, or the outcomes that are odd numbers, are: 1, 3, 5. So we have three desired outcomes.

Step 2: Apply the Probability Formula

The formula for probability is \( \frac{\text{number of desired outcomes}}{\text{total number of outcomes}} \). Substituting the values, we get \( \frac{3}{6} \).

Step 3: Simplify the Fraction

The fraction \( \frac{3}{6} \) can be simplified to \( \frac{1}{2} \). That means the probability of rolling an odd number is \( \frac{1}{2} \).

Key Concepts

Probability Theory

When we speak of probability theory, we're delving into the mathematical study of randomness and uncertainty. It's a segment of mathematics that deals with calculating the likelihood of various events occurring. In the case of rolling a die to get an odd number, it's important to understand that each of its six sides represents a unique outcome and has an equal chance of landing face up.

Under probability theory, to find the probability of a specific event, one must count the number of ways that event can occur and divide it by the total number of possible outcomes. This brings us to the concept of simple probability, which employs basic principles to solve problems like the one posed in the exercise.

Simple Probability

Simple probability is essentially the chance or likelihood of a single event occurring and is calculated using a straightforward formula. The formula for simple probability is \( P(E)=\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \). In the context of our exercise, rolling an odd number on a six-faced die, we simply count how many sides of the die are odd (1, 3, and 5), which gives us three favorable outcomes. Since there are six possible outcomes in total (one for each side of the die), the probability of rolling an odd number is \( P(\text{odd})=\frac{3}{6} \), which simplifies to \( \frac{1}{2} \).

This tells us there's a 'fifty-fifty' chance to roll an odd number, meaning if you were to roll the die many times, about half of the rolls are expected to result in an odd number.

College Algebra

College algebra forms the foundational bedrock for many mathematical concepts, including probability. It's where students hone the skill of manipulating equations and fractions — skills that are essential to solve the die probability problem.

To illustrate, in our exercise, we simplified the fraction \( \frac{3}{6} \) to \( \frac{1}{2} \). This process relies on understanding that a fraction represents division and that fractions can be reduced by finding common factors. In this case, both 3 and 6 can be divided by 3. College algebra provides the tools we need to analyze and simplify mathematical expressions, which is precisely what we do when we simplify the fraction that represents our probability. It's also important to recognize that the ability to manipulate and simplify expressions is a skill frequently exercised in the study of algebra.