Problem 15 - Algebra and Trigonometry

Question

A red die and a white die are rolled, and the numbers showing are recorded. How many different outcomes are possible? (The singular form of the word dice is die.)

Step-by-Step Solution

Answer

There are 36 different possible outcomes when rolling a red die and a white die.

Step 1: Understanding the Problem

When rolling two dice, each die is independent of the other. A red die and a white die can each land on one of six sides, showing a number between 1 and 6. We need to determine all possible combinations of outcomes from rolling both dice together.

Step 2: Counting Outcomes for One Die

Each die can land on any of its 6 faces, so it has 6 possible outcomes. The possible outcomes for one die are: 1, 2, 3, 4, 5, and 6.

Step 3: Combining Outcomes for Two Dice

Since the outcomes of the two dice are independent, each outcome from the red die can be combined with each outcome from the white die.This means we multiply the number of outcomes for the red die (6) by the number of outcomes for the white die (6).Mathematically, this is expressed as: \[ 6 \text{ (red die outcomes) } \times 6 \text{ (white die outcomes) } = 36 \text{ total outcomes} \]

Step 4: Listing Possible Outcomes

To ensure a comprehensive understanding, consider listing a few combinations: (1,1), (1,2), (1,3), ..., (1,6), (2,1), (2,2), ..., (2,6), and so on, until (6,6). There are indeed 36 combinations, as calculated previously.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. In our exercise, we use combinatorics to count the different possible outcomes when rolling two dice. Each die can show a number between 1 and 6. Instead of listing each possible outcome, combinatorics allows us to use simple multiplication to determine the total number of combinations. This makes it efficient to find all outcomes without actually enumerating every possibility.

• Arrangements: We are arranging two numbers, one from the red die and one from the white die, together in a pair like (3,5).
• Counting Principle: When you have multiple independent choices, you multiply the number of options for each to get the total number of combinations. Here it's 6 options for the red die and 6 for the white die, so 6 × 6 = 36.

Combinatorics simplifies the headache of manual counting, providing a quick way to answer "how many?" in various scenarios.

Independent Events

Understanding independent events is crucial in probability. Independent events mean that the occurrence of one event does not affect the occurrence of another. In the case of throwing two dice, the result of rolling the red die does not influence the outcome of the white die.

• Definition: Two events A and B are independent if the probability of A occurring does not change the probability of B occurring.
• Probability Rule: If A and B are independent, then the probability of both events happening is the product of their individual probabilities.

In our dice example, this independence allows us to confidently multiply the outcome probabilities of each die to find the total number of possible outcomes.

Dice Outcomes

Dice outcomes are straightforward yet essential to understand when studying probability. A standard die has six faces, numbered from 1 to 6. When you roll a die, each number has an equal chance of appearing. This concept is pivotal when dealing with problems involving dice.

• Faces on a Die: A regular die has 6 faces, meaning there are 6 potential outcomes each time you roll it.
• Fair Die: A die is fair if each outcome (1, 2, 3, 4, 5, and 6) has the same likelihood of occurring. This is crucial for unbiased probability calculations.

By understanding that each face is equally likely, we can easily compute the total outcomes for multiple dice by multiplying possibilities. This clarity ensures accurate probability assessments in games and statistical analyses.