Problem 40 - College Algebra

Question

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum between 6 and 9, inclusive.

Step-by-Step Solution

Answer

The probability is \(\frac{5}{9}\).

Step 1: Determine Total Outcomes

When rolling two six-sided dice, there are 6 possible outcomes on the first die and 6 possible outcomes on the second die. Thus, the total number of possible outcomes when rolling two dice is: \(6 \times 6 = 36\).

Step 2: List Possible Successful Outcomes

To find the successful outcomes where the sum is between 6 and 9 inclusive, we'll list the combinations: - Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1) gives 5 combinations. - Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) gives 6 combinations. - Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) gives 5 combinations. - Sum of 9: (3,6), (4,5), (5,4), (6,3) gives 4 combinations.

Step 3: Count Successful Outcomes

Add together the number of combinations for each possible sum that meets the criteria: \(5\) (for sum of 6) + \(6\) (for sum of 7) + \(5\) (for sum of 8) + \(4\) (for sum of 9) = \(20\) successful outcomes.

Step 4: Calculate the Probability

The probability of rolling a sum between 6 and 9 is the ratio of successful outcomes to total possible outcomes. Thus, the probability is: \(\frac{20}{36}\), which simplifies to \(\frac{5}{9}\).

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that studies the counting, arrangement, and combination of objects. In the context of rolling dice, combinatorics helps us understand how to determine the number of possible outcomes. It is a powerful tool for calculating probabilities as it provides methods to systematically count occurrences without having to list every possibility. For example, when rolling two dice, combinatorics allows us to easily determine that there are 36 potential combinations, each representing a unique outcome or pair of numbers.

Outcomes

Outcomes in probability refer to the possible results that occur from an experiment or trial. When two dice are rolled, each die has 6 faces, leading to a total of 36 outcomes since 6 (faces on the first die) multiplied by 6 (faces on the second die) equals 36. These outcomes represent all possible results of this sample space which is crucial to analyze when determining probabilities of certain events. In our scenario, these events are achieving a sum between 6 and 9.

Dice

Dice are small, typically cube-shaped objects with numbers on each side, commonly used in games to generate random outcomes. The most familiar type of die is the standard 6-sided die (also called a 6-sided cube), where each face displays a different number from 1 to 6. Understanding dice and their properties is fundamental in probability calculations involving dice. Each roll of a die is an independent event, meaning the outcome does not affect the outcome of the next roll, which is a key concept in probability.

Sums

Sums in dice exercises refer to the total when the numbers on the top face of the dice are added together. In problems involving sums, like finding the probability of rolling specific sums, listing all combinations becomes important to understand which results fall within the desired range.
For instance:

• A sum of 6 can be made by the combinations: (1,5), (2,4), (3,3), (4,2), (5,1).
• A sum of 7 includes combinations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
• Similarly, other sums like 8 or 9 have their unique combinations.

Comprehending sums helps us track which outcomes are favorable in calculating specific probabilities, thus making it a central concept for scenarios involving dice and desired results.