Problem 100 - Algebra and Trigonometry

Question

The figure shows that when a die is rolled, there are six equally likely outcomes: 1, 2, 3, 4, 5, or 6. What fraction of the outcomes is not less than 5?

Step-by-Step Solution

Answer

The fraction of outcomes not less than 5 when a die is rolled is \(\frac{2}{6}\) or in reduced form, \(\frac{1}{3}\).

Step 1: Identify the Outcomes

Identify the possible outcomes when a die is rolled. It could be any number from 1 to 6 inclusive. Since we are looking for outcomes that are not less than 5, we are interested in the numbers 5 and 6.

Step 2: Count the Desired Outcomes

Count the number of outcomes that are not less than 5. These numbers are 5 and 6, so there are 2 such outcomes in total.

Step 3: Calculate the Fraction

Express the probability as a fraction. The numerator is the number of outcomes that are not less than 5, which is 2. The denominator is the total number of possible outcomes when a die is rolled, which is 6. Thus, the fraction is \(\frac{2}{6}\).

Key Concepts

Understanding Probability Theory

Probability theory is a mathematical framework for quantifying the likelihood of various outcomes. It is the foundation for predicting events in numerous fields, from statistics and economics to science and engineering. When you roll a die, you're creating an experiment with a fixed number of equally possible results, and probability theory offers a way to measure the chance that a particular outcome will occur.

For example, when asked what fraction of the outcomes is not less than 5 when rolling a single die, you're using probability theory to determine how often you can expect to see a 5 or 6, compared to the entire set of possible outcomes, which are 1, 2, 3, 4, 5, and 6.

Deciphering Mathematical Outcomes

In the realm of mathematics, an outcome is the result of an experiment or activity. When rolling a die, each number that can appear—1, 2, 3, 4, 5, or 6—is a valid outcome. Identifying the outcomes that fulfill certain conditions is particularly important in probability tasks. A key step towards solving such problems is to enumerate potential outcomes, often represented as a sample space. In this case, by focusing on the outcomes not less than 5, one narrows down to two specific results: 5 and 6. Understanding how to isolate these outcomes based on the given conditions is essential in analyzing probability experiments.

Furthermore, the distinction between 'at least' and 'more than' requires close attention. Outcomes 'not less than 5' include the number 5 itself, along with the number 6, because both meet the criteria established by the question.

Mastering Fraction Calculation

Fraction calculation is an integral part of probability assignments, as it presents probability in the simplest form. When we express probability as a fraction, the numerator represents the count of favorable outcomes (in this case, outcomes not less than 5), while the denominator indicates the total number of possible outcomes.

In the given die-rolling problem, the fraction is initially calculated as \(\frac{2}{6}\), with 2 representing the two favorable outcomes (5 and 6), and 6 symbolizing the complete set of potential outcomes when tossing a die. Simplifying the fraction is crucial for clear understanding: dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case, you obtain \(\frac{1}{3}\). Simplification makes the fraction more comprehensible and is a necessary skill in mathematical practice. Remember, the simplified fraction \(\frac{1}{3}\) communicates the same probability as \(\frac{2}{6}\), but in a more straightforward manner.