Marbles Probability Calculating The Chance Of Not Red

This article delves into the realm of probability, specifically focusing on a scenario involving marbles of different colors in a bag. We will explore how to calculate the probability of selecting a marble that is not red, a common type of probability question encountered in mathematics. The problem presented is a classic example of basic probability, which involves determining the likelihood of a specific event occurring. In this case, the event is drawing a marble that isn't red from a bag containing green, yellow, and red marbles. Probability is a fundamental concept in mathematics that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The basic formula for probability is:

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

Understanding this formula is the key to solving a wide range of probability problems. The 'number of favorable outcomes' refers to the number of outcomes that align with the event you're interested in, while the 'total number of possible outcomes' represents the entire set of outcomes that could occur. Let's apply this understanding to the specific problem at hand, where Marsha has a bag filled with marbles of different colors, and we want to find the probability of picking a marble that isn't red. By carefully identifying the favorable outcomes and the total possible outcomes, we can accurately calculate the probability and gain a deeper appreciation for this core mathematical concept.

Problem Statement

Marsha has a bag containing marbles of three different colors: green, yellow, and red. The number of marbles of each color is as follows:

• 4 green marbles
• 8 yellow marbles
• 20 red marbles

The question we aim to answer is: If Marsha chooses one marble from the bag at random, what is the probability that the marble she chooses is not red?

This problem requires us to apply the fundamental principles of probability. We need to determine the total number of marbles in the bag, identify the number of marbles that are not red, and then use these values to calculate the probability. The process involves several key steps, which we will break down in detail to ensure a clear and comprehensive understanding. First, we need to calculate the total number of marbles, which will serve as the denominator in our probability calculation. This involves summing the number of marbles of each color. Next, we need to determine the number of marbles that are not red, which will be the numerator in our probability calculation. This involves adding the number of green and yellow marbles. Finally, we will divide the number of non-red marbles by the total number of marbles to obtain the probability. By following these steps carefully, we can arrive at the correct answer and gain a better grasp of how probability works in practice. The concept of 'not red' is crucial here. Instead of directly calculating the probability of picking a red marble, we're looking for the probability of the complementary event – picking a marble that is either green or yellow. This type of problem highlights the importance of understanding complementary probabilities, which can often simplify complex calculations.

Solution

Step 1: Calculate the Total Number of Marbles

To find the total number of marbles, we add the number of marbles of each color:

$$Total = Green + Yellow + Red$$

$$Total = 4 + 8 + 20$$

$$Total = 32$$

Therefore, there are a total of 32 marbles in the bag. This total number of marbles represents the total possible outcomes when Marsha chooses one marble from the bag. Understanding the total number of outcomes is crucial for calculating probability, as it forms the denominator in the probability fraction. Without knowing the total possible outcomes, we cannot accurately determine the likelihood of a specific event occurring. In this case, the total number of marbles serves as the foundation for calculating the probability of Marsha choosing a marble that is not red. By correctly calculating the total number of marbles, we ensure that our subsequent probability calculations are accurate and reliable. This step is a fundamental part of solving probability problems, and it is essential to approach it with care and precision. The total number of marbles, 32, will be used later to divide the number of favorable outcomes, giving us the final probability. Make sure to double-check this calculation to avoid any errors that could affect the final result. A common mistake is to overlook one of the colors or to miscalculate the sum. Double-checking your work here can save you from making mistakes later on in the solution. Now that we have the total number of marbles, we can move on to the next step, which involves determining the number of marbles that are not red.

Step 2: Calculate the Number of Marbles That Are Not Red

To find the number of marbles that are not red, we add the number of green and yellow marbles:

$$NonRed = Green + Yellow$$

$$NonRed = 4 + 8$$

$$NonRed = 12$$

Thus, there are 12 marbles in the bag that are not red. These 12 marbles represent the favorable outcomes for the event we're interested in – choosing a marble that isn't red. The concept of favorable outcomes is central to probability calculations. It refers to the number of outcomes that align with the specific event we are trying to determine the likelihood of. In this case, the favorable outcomes are the green and yellow marbles because they satisfy the condition of being 'not red'. By correctly identifying the number of favorable outcomes, we can accurately calculate the probability of the event occurring. This step requires a careful understanding of the problem statement and the event we are trying to analyze. Misinterpreting the event can lead to an incorrect calculation of the number of favorable outcomes, which in turn will result in an incorrect probability. In this case, the phrase 'not red' is the key, and we must ensure that we are only counting the marbles that are green or yellow. Now that we know the number of non-red marbles, we can proceed to the final step, which involves calculating the probability by dividing the number of favorable outcomes by the total number of possible outcomes. This step will give us the probability of Marsha choosing a marble that is not red, which is the ultimate goal of this problem.

Step 3: Calculate the Probability of Choosing a Marble That Is Not Red

Now, we can calculate the probability using the formula:

$$P(NonRed) = \frac{NonRed}{Total}$$

$$P(NonRed) = \frac{12}{32}$$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

$$P(NonRed) = \frac{12 ÷ 4}{32 ÷ 4}$$

$$P(NonRed) = \frac{3}{8}$$

Therefore, the probability that the marble Marsha chooses is not red is $\frac{3}{8}$. This probability represents the likelihood of the event occurring – in this case, Marsha picking a marble that is either green or yellow. The probability is expressed as a fraction, where the numerator represents the number of favorable outcomes (non-red marbles) and the denominator represents the total number of possible outcomes (total marbles). Simplifying the fraction is a crucial step in presenting the probability in its simplest form, which makes it easier to understand and compare with other probabilities. In this case, simplifying the fraction $\frac{12}{32}$ to $\frac{3}{8}$ provides a clearer representation of the probability. The simplified fraction tells us that for every 8 marbles in the bag, 3 of them are not red. This gives us a proportional understanding of the likelihood of picking a non-red marble. Probability calculations are used in a wide variety of real-world applications, from predicting weather patterns to assessing financial risks. Understanding how to calculate probability is a valuable skill that can help us make informed decisions in many aspects of life. By carefully following the steps outlined in this solution, we have successfully calculated the probability of Marsha choosing a marble that is not red. The final answer, $\frac{3}{8}$, represents the solution to the problem and demonstrates the application of basic probability principles.

Final Answer

The probability that the marble Marsha chooses is not red is $\frac{3}{8}$, which corresponds to option C.

Conclusion

In this problem, we successfully calculated the probability of Marsha choosing a marble that is not red from a bag containing green, yellow, and red marbles. We followed a step-by-step approach, which included calculating the total number of marbles, determining the number of marbles that are not red, and then applying the probability formula. This process demonstrates the fundamental principles of probability and how they can be applied to solve real-world problems. Understanding probability is essential in various fields, including mathematics, statistics, and decision-making. By working through this problem, we have reinforced our understanding of these principles and developed our problem-solving skills. The key takeaways from this problem include the importance of identifying the total possible outcomes, determining the favorable outcomes, and then applying the probability formula. We also learned the significance of simplifying fractions to express probabilities in their simplest form. The concept of complementary events, such as 'not red' in this case, is also an important aspect of probability that we have explored. By mastering these concepts, we can confidently tackle a wide range of probability problems. Probability problems can often seem daunting at first, but by breaking them down into smaller, manageable steps, we can approach them with confidence and arrive at the correct solution. This problem serves as a valuable example of how to apply basic probability principles and can be used as a foundation for understanding more complex probability concepts in the future. Remember to always double-check your calculations and ensure that you have correctly identified the total possible outcomes and the favorable outcomes. With practice, you can become proficient in solving probability problems and appreciate their relevance in various real-world scenarios. The ability to calculate and interpret probabilities is a valuable skill that can help you make informed decisions and understand the likelihood of different events occurring.