Problem 24 - College Algebra with Modeling and Visualization

Question

A red die and a blue die are thrown. How many ways are there for both dice to show an even number?

Step-by-Step Solution

Answer

There are 9 ways for both dice to show an even number.

Step 1: Identify Possible Outcomes for One Die

Each die, when rolled, can show one of six numbers: 1, 2, 3, 4, 5, or 6. We are tasked with only counting the outcomes where the result is an even number. The even numbers on a die are 2, 4, and 6.

Step 2: Determine Possible Ways for One Die

Since a single die can show three even numbers, there are 3 possible outcomes for each die separately being even. This calculation is independent for each die, be it red or blue.

Step 3: Calculate Total Combinations

To find the total number of ways both dice show an even number, multiply the number of options for an even number from the red die by the number from the blue die. So, we calculate: \( 3 \times 3 = 9 \) This means there are 9 combinations where both dice show an even number.

Key Concepts

Combinatorics

Combinatorics is a key branch of mathematics that focuses on counting, arranging, and combining objects. It is crucial for solving probability problems, such as determining possible outcomes for dice rolls.
A typical use of combinatorics is to find how many different ways we can arrange a set of items. In the context of dice, each die has specific outcomes it can show.

• For a single die, there are 6 possible results: 1, 2, 3, 4, 5, and 6.
• If we focus on even numbers, the options reduce to 3: 2, 4, and 6.

Understanding combinatorics helps us compute combinations of events like the number of different outcomes when rolling two separate dice. By applying the principle of multiplying the possible outcomes of one event by another, we easily calculate complex probabilities.

Independent Events

An independent event is one where the outcome of one event does not affect the outcome of another. In the context of rolling dice, this concept plays a critical role.
Two dice rolls are independent because the number one die shows does not influence the number the other die will display. Hence, each die acts independently.

• This means that the probability of each die showing an even number remains the same, regardless of what the other die shows.
• For instance, the probability of a red die showing an even number remains at any one time 3 out of 6 or \( \frac{1}{2} \).

These independent probabilities allow us to calculate the total probability of both dice showing an even number by multiplying their individual probabilities.

Dice Outcomes

When exploring dice outcomes, it's essential to examine the possible results from each roll. With a standard die, you have 6 faces, each showing a number from 1 to 6.
In probability exercises, we often categorize these outcomes based on certain criteria, like even numbers.

• The even numbers on a die include 2, 4, and 6, making up half of the total outcomes.
• Given two dice are involved, each with 3 even options, we can calculate possible outcomes by \( 3 \text{ (from red die)} \times 3 \text{ (from blue die)} = 9 \) total ways.

Understanding how to efficiently list and count these outcomes is crucial for solving many problems in probability and ensuring the accurate application of probability principles.