Problem 97 - College Algebra

Question

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is even or greater than \(3 ?\)

Step-by-Step Solution

Answer

The fraction of outcomes that are even or greater than 3 is \(\frac{2}{3}\).

Step 1: Identify the total number of outcomes

A six-sided die has 6 possible outcomes: \(1, 2, 3, 4, 5, \) and \(6\). So the total number of outcomes (\(n(S)\)) is \(6\).

Step 2: Identify the fitting outcomes

We need to find outcomes that are either 'even' or 'greater than 3'. The even numbers on the die are \(2, 4,\) and \(6\), and the numbers greater than 3 are \(4, 5,\) and \(6\). So, the fitting outcomes are \(2, 4, 5, \)and \(6\). Therefore, the number of fitting outcomes (\(n(E)\)) is \(4\).

Step 3: Calculate the probability

The probability or the fraction of the outcomes that are 'even' or 'greater than 3' is given by the ratio of the number of fitting outcomes (\(n(E)\)) to the total number of outcomes (\(n(S)\)): \( P(E) = \frac{n(E)}{n(S)} = \frac{4}{6} = \frac{2}{3}\).

Key Concepts

Even Outcomes in Probability

When considering events like rolling a die, each side landing face up is one possible outcome. In the realm of probability, we often categorize these outcomes based on certain characteristics they may have. One such characteristic is 'evenness'—whether the outcome number is divisible by two without any remainder.

For a six-sided die, the 'even' outcomes are 2, 4, and 6. Since each roll of the die is independent, we assume each outcome is equally likely. So, when asked about the probability of rolling an even number, we are looking at how many of the total outcomes (2, 4, 6) fit that criterion versus the total number of possible outcomes (1 through 6). In essence, even outcomes in probability hinge on identifying specific members of a set that share a common trait—in this case, evenness—and comparing their frequency to all possible outcomes.

Calculating Probability

Probability expresses the likelihood of an event happening and is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. This is represented by the formula:\[ P(E) = \frac{n(E)}{n(S)} \]where \( P(E) \) is the probability of the event E occurring, \( n(E) \) is the number of favorable outcomes, and \( n(S) \) is the total number of possible outcomes. When calculating the probability for the exercise given, we first count the total number of possible outcomes when rolling a die, which is 6. Then we count the number of outcomes that meet our favorable condition—in this case, being even or greater than 3. Finally, we divide the number of favorable outcomes by the total number of outcomes to find our probability fraction. Understanding this formula and how to apply it is foundational to solving probability problems.

Probability Fraction

In probability, we often express the likelihood of an event as a fraction. This fraction, known as the probability fraction, is the mathematical representation of the chance that a particular event will occur. The numerator of the fraction indicates the count of favorable outcomes, while the denominator corresponds to the total number of possible outcomes.

In the given exercise, the fraction \( \frac{4}{6} \) represents the probability of rolling either an even number or a number greater than 3 on a die, which simplifies to \( \frac{2}{3} \) after reducing the fraction. This simplification is an integral part of presenting the probability in its simplest form. By conveying probabilities as reduced fractions, we offer a clear and concise depiction of the chances involved, which is much easier for students to interpret and understand.