Problem 63 - Algebra for College Students

Question

What are the odds in favor of getting a sum of 5 with one toss of a pair of dice? 1 to 8

Step-by-Step Solution

Answer

The odds in favor are 1 to 8.

Step 1: Understand the problem

The exercise asks for the odds in favor of getting a sum of 5 when rolling two dice. We need to understand how many ways we can achieve this sum and compare it to the total ways to roll two dice.

Step 2: Identify total outcomes

Rolling a pair of dice results in a combination of two numbers from 1 to 6. Therefore, the total number of possible outcomes is calculated as follows: \[ 6 \times 6 = 36 \] This means there are 36 possible outcomes.

Step 3: Identify successful outcomes for a sum of 5

List the combinations of die faces that add up to 5, which are: (1,4), (2,3), (3,2), (4,1). Each pair represents one possible successful outcome.

Step 4: Count successful outcomes

There are 4 successful combinations that result in a sum of 5: - (1,4) - (2,3) - (3,2) - (4,1). Thus, there are 4 successful outcomes.

Step 5: Calculate odds in favor

The odds in favor of an event are calculated as the ratio of successful outcomes to unsuccessful outcomes. We have 4 successful outcomes and 36 total outcomes, so the number of unsuccessful outcomes is: \[ 36 - 4 = 32 \].The odds in favor of rolling a sum of 5 are: \[ \text{Odds in favor} = \frac{4}{32} \].This can be simplified to: \[ \frac{1}{8} \].

Key Concepts

Odds Calculation

Calculating odds is an important concept when dealing with probability, especially in scenarios involving dice rolls. Odds in favor of an event happening are calculated by comparing the number of successful outcomes to the number of unsuccessful outcomes. For instance, when you roll two dice, each die can land on a number between 1 and 6, meaning there are 36 total possible outcomes. If you want to roll a sum of 5, first identify all the successful outcomes that meet this criterion. Once you've done that, count all remaining outcomes as unsuccessful. Finally, calculate the odds by forming a ratio of the successful outcomes over the unsuccessful outcomes. For the given exercise, there are 4 successful outcomes that result in a sum of 5. There are 32 unsuccessful outcomes when rolling two dice (36 total outcomes minus the 4 successful ones). Therefore, the odds in favor of rolling a sum of 5 are \( \frac{4}{32} \), which simplifies to \( \frac{1}{8} \). This is a useful ratio because it directly tells you how likely you are to succeed in gaining that sum, compared to not achieving it.

Successful Outcomes

Successful outcomes are the specific results that meet the criteria you are looking for when rolling dice. In our exercise, a successful outcome is any combination of dice that adds up to a sum of 5 when rolled together. It's important to carefully determine what makes an outcome successful based on the problem at hand. For dice rolls, to find successful outcomes for a sum of 5, you list the dice combinations that achieve this sum. The possible pairs are: (1,4), (2,3), (3,2), and (4,1). Each of these pairs counts as one successful outcome, as they all add up to 5. When solving probability exercises, listing successful outcomes helps in calculating odds or probabilities because they form the numerator in related computations. The more successful outcomes you have, the higher your chances are of matching the desired result from the probability exercise.

Total Outcomes

When rolling two six-sided dice, it's essential to calculate the total possible outcomes to understand the probability of any result. Each die has 6 sides, offering numbers from 1 through 6. Therefore, when you roll both dice together, you can calculate the total number of outcomes using simple multiplication. The formula is: \[ 6 \times 6 = 36 \]This implies that each roll could result in any of 36 different combinations. This total count is crucial because it serves as the denominator in probability and odds calculations. Knowing the total outcomes allows you to measure the likelihood of specific results, like rolling a sum of 5. Understanding this concept helps you conceptualize the odds and analyze different dice-related games or scenarios.

Dice Combinations

When dealing with dice, comprehending the various combinations that can result is key to understanding probability. Each die in a pair has independent outcomes, meaning the result on one does not affect the other. This independence creates a straightforward way of finding combinations. For example, each die can show numbers 1 through 6. To determine all possible dice combinations, consider the interplay between both dice. Pair the result of one die with each result on the other, forming the set of all possible pairs. This gives us combinations such as (1,1), (1,2), up to (6,6). In total, there are 36 possible pairs when rolling two dice. Recognizing which of these combinations result in a desirable outcome, such as the combined roll totaling 5, is a fundamental skill in problems related to probability with dice. By exploring these combinations, you can confidently calculate both successful and unsuccessful outcomes, ultimately assisting in deriving odds and probabilities.