Problem 32 - Algebra 2

Question

A die is rolled, Find each probability. \(P(\text { even })\)

Step-by-Step Solution

Answer

The probability of rolling an even number is \( \frac{1}{2} \).

Step 1: Identify Total Possible Outcomes

When rolling a standard six-sided die, each side has an equal chance of landing face up. Thus, the total number of possible outcomes is 6.

Step 2: Identify Favorable Outcomes

The favorable outcomes are the ones that result in even numbers. On a six-sided die, these numbers are 2, 4, and 6.

Step 3: Count the Favorable Outcomes

There are three even numbers on a die: 2, 4, and 6. So, there are 3 favorable outcomes.

Step 4: Calculate the Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes: \( P(\text{even}) = \frac{\text{Number of Even Outcomes}}{\text{Total Number of Outcomes}} = \frac{3}{6} \).

Step 5: Simplify the Probability

Simplify the fraction: \( \frac{3}{6} = \frac{1}{2} \). Therefore, the probability of rolling an even number is \( \frac{1}{2} \).

Key Concepts

Dice

A die is a small cube used in various games. It has six faces, each marked with a different number of dots from 1 to 6.
Dice are used to generate random numbers since each side has an equal probability of landing face-up. This makes dice an excellent tool for understanding probability, as each roll is an independent event.
With a single roll, the dice can yield any one of six outcomes, making it a simple yet effective way to study basic probability concepts.

• Each side represents a possible outcome.
• Equal chances ensure fairness in games.

Understanding the nature of dice and their outcomes lays the foundation for grasping more complex probability concepts later on.

Even numbers

Even numbers are integers divisible by 2 without leaving a remainder. On a standard die, the even numbers are 2, 4, and 6.
These numbers are important in many probability problems, especially when calculating probabilities related to rolling dice. Here’s why they matter:

• They represent half of the possible outcomes on a die.
• Even numbers are evenly spaced apart, making calculations straightforward.

In dice probability, knowing the count and position of even numbers helps in determining outcomes related to them. Remember, the sequence of even numbers on a die is regular, ensuring predictability.

Outcomes

In probability, an outcome is a possible result that can occur from an experiment or action. When you roll a die, there are 6 possible outcomes:

• 1
• 2
• 3
• 4
• 5
• 6

Each number represents an individual outcome. When considering probabilities, the total possible outcomes establish the foundation for calculating chances.
Understanding outcomes is crucial because:

• It helps in visualizing the complete scenario of an event.
• Allows for identifying which outcomes are favorable for a specific event.

This knowledge allows students to compute probabilities by comparing favorable outcomes against the total outcomes.

Fraction simplification

Fraction simplification is a process used to make a fraction easier to understand by making it smaller or simplified, without changing its value.
This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
In the context of probability, simplifying fractions makes it easier to interpret odds. For instance, when rolling a die, the probability of rolling an even number starts as \( \frac{3}{6} \).

• Here, the GCD of 3 and 6 is 3.
• By dividing both the numerator and denominator by 3, we simplify \( \frac{3}{6} \) to \( \frac{1}{2} \).

This simplified fraction means that there is a 1 in 2 chance, or 50%, probability of rolling an even number. Simplification not only makes the fraction easier to read but also assists in understanding the probability in real-world contexts.