Problem 17 - Algebra for College Students

Question

What is the probability of getting a sum of 8 with one roll of a pair of dice? \(\frac{5}{36}\)

Step-by-Step Solution

Answer

The probability is \( \frac{5}{36} \).

Step 1: Understanding the Problem

We need to determine how many outcomes of rolling two dice result in a sum of 8. First, recognize that each die has 6 sides, numbered 1 through 6.

Step 2: Identifying Relevant Outcomes

Consider the pairs of numbers from two dice that can add up to 8: (2,6), (3,5), (4,4), (5,3), and (6,2). These are the only combinations because both dice must show whole number values from 1 to 6.

Step 3: Counting Total Possible Outcomes

In a dice roll, each die has 6 potential outcomes. Therefore, for two dice, the total number of possible outcomes is the product: 6 (outcomes for the first die) × 6 (outcomes for the second die) = 36 total outcomes.

Step 4: Calculating the Probability

We found that there are 5 successful outcomes that result in a sum of 8. Apply the formula for probability of an event: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). In this case, \( P(8) = \frac{5}{36} \).

Step 5: Confirming the Solution

The step-by-step calculations confirm that there are indeed 5 outcomes that result in a sum of 8, and the total number of possible outcomes is 36, giving us a probability of \( \frac{5}{36} \).

Key Concepts

Understanding Dice Probability

When you roll a pair of dice, dice probability refers to figuring out the chance of a particular outcome or sum showing up. Each die has 6 faces, and when rolled together, there's a multitude of combinations you might see. This involves understanding that each die's outcome is independent of the other, meaning what one die shows doesn't affect what the other die will display.

• The number of faces on a die is 6.
• Rolling two dice produces outcomes like (1,1), (1,2), ..., (6,6).
• Each side has an equal chance of appearing.

Understanding these basic components helps in calculating probabilities, such as the likelihood of a particular sum, like 8, showing up.

Exploring the Sum of Dice

The sum of dice relates to the total number you get after adding the numbers shown on each die. With two six-sided dice, the smallest sum is 2 (1+1), while the largest is 12 (6+6). Knowing the range of possible sums is critical for calculating their probabilities.
Each possible outcome from a dice roll pair, such as (2,6) or (3,5), results in different sums:

• The sum of 2 only occurs with (1, 1).
• The sum of 3 can come from (1,2) or (2,1).
• The sum of 8, which interests us here, can be obtained from several combinations, such as (2,6) or (4,4).

Understanding these combinations helps you see how often a particular sum appears among the dice rolls.

Mastering Probability Calculation

Probability calculation in dice involves determining how likely a certain sum or outcome is to happen based on the number of successful results divided by the total possible results. This is often expressed as a fraction.
The formula for probability, expressed as \( P \) is:

• \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

For our problem, where we seek the probability of the sum being 8, first identify the favorable results (like rolling a 2 and 6). Then, use the formula to compute:

• Total outcomes with two dice: 6 sides × 6 sides = 36 possible outcomes.
• Outcomes that produce a sum of 8: There are 5 combinations, such as (2,6) and (4,4).
• Probability of rolling a sum of 8: \( \frac{5}{36} \).

Calculating dice probabilities requires meticulously counting viable combinations and comparing them to the total possibilities.

Recognizing Favorable Outcomes

To find favorable outcomes, look for specific die combinations that yield the desired sum. In probability, favorable outcomes are those results that satisfy the condition we're examining.
For the sum of 8 using two dice, the combinations are:

• (2,6) and (6,2)
• (3,5) and (5,3)
• (4,4)

Together these make 5 favorable outcomes. Each combination highlights different ways the two dice can add up to the target sum. Recognizing these combinations is essential to understanding and solving probability questions effectively. This identification process is crucial for determining probabilities accurately.