Question

Two ordinary dice are rolled. In how many different ways can they fall? How many of these ways will give a sum of nine?

Short answer

There are 36 different ways two ordinary dice can fall. Out of these, 4 combinations will give a sum of 9.

Step-by-step solution

Calculate Total Number of Possible Outcomes

To calculate the total number of possible outcomes, we need to multiply the number of faces on each die. Since each die has 6 faces, the total number of outcomes can be found by: \( Total Outcomes = Die1 \times Die2 = 6 \times 6 = 36 \)

List All Outcomes With a Sum of 9

Now, let us list all possible combinations where the sum of the outcomes on the two dice is equal to 9: - Die 1 shows 3, Die 2 shows 6: 3 + 6 = 9 - Die 1 shows 4, Die 2 shows 5: 4 + 5 = 9 - Die 1 shows 5, Die 2 shows 4: 5 + 4 = 9 - Die 1 shows 6, Die 2 shows 3: 6 + 3 = 9 So, there are 4 combinations producing a sum of 9.

Conclusion

In conclusion, there are a total of 36 possible outcomes when rolling two ordinary dice. Out of these, 4 combinations will produce a sum of 9.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that involves counting, arranging, and analyzing combinatorial structures. It helps us determine the number of ways events or objects can be arranged or combined. When applying combinatorics to the problem of rolling two dice, we can use basic counting methods to figure out how many possible outcomes exist. This field arms us with the tools to not only count outcomes but also to make sense of patterns and structures in combinations. When using combinatorics in practical scenarios like dice rolls, you can greatly simplify problems by breaking down each step into more manageable parts.

Dice Roll Outcomes

When we talk about dice roll outcomes, we're dealing with all the possible results that can occur when dice are rolled. For a single six-sided die, there are six outcomes, ranging from 1 to 6. With two dice, each die can land on any of these six faces, which expands the set of possible outcomes. To find the total number of possible outcomes for two six-sided dice, multiply the number of outcomes for each die:

• Total outcomes = 6 (from the first die) × 6 (from the second die) = 36 outcomes.

Understanding how to calculate the number of outcomes lays the groundwork for deeper explorations into probabilities and likelihoods of specific results.

Sum of Two Dice

Calculating the sum of two dice involves finding all the combinations of dice faces that add up to a specific number. When looking for a sum of 9 with two dice, you need to consider each die's result:

• Die 1 shows 3, Die 2 shows 6: 3 + 6 = 9
• Die 1 shows 4, Die 2 shows 5: 4 + 5 = 9
• Die 1 shows 5, Die 2 shows 4: 5 + 4 = 9
• Die 1 shows 6, Die 2 shows 3: 6 + 3 = 9

These calculations show there are four ways to get a sum of 9 with two dice. Finding these combinations is a common application of both combinatorics and probability.

Basic Counting Principles

Basic counting principles are essential for solving probability problems. When two actions are independent (like rolling two dice), you can determine the total number of outcomes by multiplying the number of possibilities for each action. This is known as the Fundamental Principle of Counting and ensures you account for all possible outcomes without omission or repetition.
In the context of the problem:

• Each die can roll 6 different numbers.
• Thus, roll of Die 1 being independent of roll of Die 2, we find that the possible outcomes = 6 × 6 = 36.

Understanding and applying these principles provides the backbone for more complex calculations and insights in probability.