Problem 17 - Algebra and Trigonometry

Question

Rolling Three Dice A red die, a blue die, and a white die are rolled, and the numbers that show are recorded. How many different outcomes are possible?

Step-by-Step Solution

Answer

216 different outcomes are possible when rolling three distinct dice.

Step 1: Understanding the Problem

We need to find out how many different outcomes are possible when a red die, a blue die, and a white die are rolled simultaneously. Each die is a standard 6-sided die with faces numbered from 1 to 6.

Step 2: Determine Possible Outcomes for One Die

Since each die is a standard 6-sided die, it has 6 possible outcomes (1 through 6).

Step 3: Calculate Total Outcomes for Three Dice

Since the outcome of each die roll is independent of the others, we use the multiplication rule to find the total number of combinations. We multiply the number of outcomes for each die: 6 (red die) × 6 (blue die) × 6 (white die).

Step 4: Application of Multiplication Rule

Using the multiplication rule, we calculate the total number of different outcomes by multiplying the individual outcomes: \[ 6 \times 6 \times 6 = 216 \]

Key Concepts

Multiplication Rule

The multiplication rule in combinatorics is a powerful tool for finding the total number of possible outcomes in a series of events. This rule applies when each event is independent, meaning the outcome of one does not affect the others. To understand it fully, imagine having a set process with several steps or stages. If each step can be completed in a certain number of ways, the multiplication rule calculates the total ways the whole process can be completed by multiplying the number of possibilities at each step.

For example, if you roll three different colored dice, each die can result in any of 6 outcomes from 1 to 6. Thus, if you roll the red die first, there are 6 possible results. Following this, the blue die also has 6 possible outcomes, and the same goes for the white die. Therefore, the multiplication rule tells us to multiply these outcomes:

• 6 outcomes for the red die
• 6 outcomes for the blue die
• 6 outcomes for the white die

Thus, the calculation becomes: \[6 \times 6 \times 6 = 216\] This means there are 216 different possible combinations when rolling three dice.

Probability

Probability is the measure of how likely an event is to occur, often quantified as a number between 0 and 1. A probability of 1 indicates certainty, whereas a probability of 0 indicates impossibility.

To find the probability of a particular outcome, you divide the number of favorable outcomes by the total number of possible outcomes. In the context of rolling dice, suppose you want to find the probability of rolling a specific combination like all dice showing a 6. Here’s how you would calculate it:

• There is only 1 way to roll a 6 on each die.
• For all three dice to show 6, the single favorable outcome is (6,6,6).
• The total number of outcomes, as calculated using the multiplication rule, is 216.

The probability is then: \[\frac{1}{216}\]This showcases how probability and combinatorics work together to assess the chance of particular occurrences within a set of possibilities.

Independent Events

Understanding independent events is crucial in probability and combinatorics. Independent events are those where the outcome of one event does not affect the outcome of another. This is a key feature when calculating probabilities via the multiplication rule.

When considering the rolling of multiple dice, each die is considered an independent event. This means whether you roll a 1 or a 6 on the red die doesn't impact what you can roll on the blue or white die. The principle of independence simplifies calculations because it allows us to multiply the number of outcomes across independent events.

For example, rolling a 4 on a red die does not influence the possibility of rolling a 3 or any other number on the blue die. This separation allows predictions and calculations to remain straightforward and manageable even in seemingly complex scenarios. By recognizing events as independent, we can apply the multiplication rule directly and ease the process of understanding large sample spaces, just like calculating outcomes when rolling multiple dice.