Problem 16 - Algebra and Trigonometry

Question

Rolling a Pair of Dice A red die and a white die are rolled, and the numbers that show are recorded. How many different out-comes are possible? (The singular form of the word dice is die.)

Step-by-Step Solution

Answer

There are 36 possible outcomes.

Step 1: Understand the Problem

We are rolling two different dice, so we need to consider each die separately. One die is red and one is white, and we're considering the numbers that appear when both are rolled.

Step 2: Determine Outcomes for One Die

Each die has 6 faces, numbered from 1 to 6. Therefore, each die can show any one of these 6 outcomes on a single roll.

Step 3: Determine Combined Outcomes

Since the two dice are independent, the total number of outcomes can be determined by multiplying the number of outcomes of the red die by the number of outcomes of the white die. So, the total number of outcomes is given by: 6 (red die) * 6 (white die) = 36.

Key Concepts

Dice Outcomes

When we think about dice, these small cubes used in many games, it's important to know they have six sides. Each face shows a number from 1 to 6.
Each number that comes up on top when you roll the dice is called an outcome.
Now, let's consider what happens when we roll two dice, one red and one white. For each die, there are six possible numbers that can appear. This means the red die can show any number from 1 to 6, and so can the white die.
To find all the outcomes when rolling both dice, we simply look at all possible combinations of the numbers that could appear on both the red and the white dice.

• The first die (red) could show a 1, paired with any of the six numbers on the white die.
• The first die (red) could show a 2, again paired with any of the six on the white die.
• This pattern repeats until the red die shows a 6, matched with every number on the white die.

By understanding this, we can see there are many possible outcomes from rolling even two simple dice.

Combinatorial Analysis

Combinatorial analysis is a nifty branch of mathematics used to calculate the number of different ways a certain event can occur. It's especially useful when dealing with dice outcomes, where each die can land in several ways.
With two independent dice, combinatorial analysis helps us figure out the combined possibilities of their outcomes.
Here's how it works: Since each die has 6 sides, the number of outcomes for one die is 6.
When calculating the possible outcomes for two dice, the rule of product applies. This rule says if one event can happen in a particular number of ways, and a second independent event can happen in another number of ways, their combined outcomes are simply the product of these two numbers.

• For the red die, there are 6 outcomes.
• For the white die, there are also 6 outcomes.
• The total number of combined outcomes is calculated as \(6 \times 6 = 36\).

Combinatorial analysis makes it easier to handle many dice scenarios by breaking down the problem into manageable parts.

Independent Events

In probability, understanding what independent events mean can be very useful. When events are independent, the outcome of one does not affect the outcome of another. In our dice example, rolling the red die doesn't change the possible outcomes for the white die.
Each die acts on its own, separate from the other.
This means the chances of a particular number showing up on each die are not influenced by what happens with the other die.
Imagine rolling two dice: the number on the red die is independent of the number on the white die. You still have the same 1 in 6 chance for a particular number on either die.

• The independence of the dice makes predicting the total outcomes straightforward: We simply multiply the outcomes of both dice: 6 for the red die times 6 for the white die.
• This independence simplifies calculations in many probability problems.

Understanding independent events is key to solving exercises where multiple dice, coins, or other random events are considered together.