Probability Of Drawing A Green Token Step By Step Solution

This article delves into the fascinating world of probability, using a classic token-drawing scenario as our example. We'll explore how to calculate the likelihood of specific events when drawing tokens from a bag, considering the changing composition of the bag after each draw. This is a fundamental concept in probability theory, and understanding it can unlock a deeper appreciation for the mathematics that governs chance and uncertainty. By focusing on a clear and structured approach, this article aims to demystify probability calculations and provide a solid foundation for tackling more complex problems.

H2: The Initial Setup: Defining the Problem

Before we dive into calculations, let's clearly define the problem. Probability is at the heart of this question. Marcus has a bag filled with tokens of different colors: 10 blue, 8 green, and 12 red. The total number of tokens in the bag is therefore 10 + 8 + 12 = 30. This total number is crucial because it forms the basis of our initial probability calculations. The problem states that Marcus randomly draws a token, and the first token drawn is green. This is our key event of interest. We need to determine the probability of this event occurring. Probability, in its simplest form, is the ratio of favorable outcomes to the total number of possible outcomes. In this case, a favorable outcome is drawing a green token, and the total possible outcomes are the total number of tokens in the bag. The initial setup is paramount as it sets the stage for all subsequent calculations. A clear understanding of the initial conditions, such as the number of tokens of each color and the total number of tokens, is essential for accurately determining probabilities. This foundation allows us to analyze the event in question and apply the principles of probability effectively. The number of green tokens, 8, is particularly important because it represents the number of favorable outcomes for our initial event. This number will be the numerator in our probability fraction. The total number of tokens, 30, represents the total possible outcomes, and it will be the denominator in our probability fraction. By understanding these basic concepts and applying them to the initial setup, we can begin to calculate the probability of drawing a green token.

H2: Calculating the Probability of Drawing a Green Token on the First Draw

Now, let's calculate the probability of Marcus drawing a green token on his first attempt. As we established earlier, probability is the ratio of favorable outcomes (drawing a green token) to the total number of possible outcomes (drawing any token). Initially, there are 8 green tokens and 30 total tokens in the bag. Therefore, the probability of drawing a green token on the first draw is 8/30. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This simplification gives us 4/15. This means that there is a 4 out of 15 chance that Marcus will draw a green token on his first try. This calculation highlights a key principle of probability: it's a measure of the likelihood of an event occurring, expressed as a fraction or a percentage. The higher the probability, the more likely the event is to occur. In this case, a probability of 4/15 suggests that drawing a green token is not a highly probable event, but it's also not extremely unlikely. The calculation also demonstrates the importance of simplifying fractions in probability. Simplifying the fraction makes it easier to understand and compare probabilities. A simplified fraction also reduces the risk of errors in subsequent calculations. The probability of 4/15 represents the likelihood of drawing a green token before any tokens have been removed from the bag. This is an important distinction, as the probability will change if tokens are removed and not replaced. This initial probability serves as a baseline for understanding how the probabilities of subsequent draws will be affected by the outcome of the first draw. Therefore, accurately calculating this initial probability is crucial for solving the overall problem and understanding the dynamics of probability in this scenario. This step-by-step calculation ensures clarity and provides a solid foundation for understanding more complex probability scenarios.

H2: The Correct Answer and Why

The correct answer to the question, “What is the probability of Marcus drawing a green token on the first draw?” is B. 4/15. We arrived at this answer by understanding the basic principles of probability and applying them to the given scenario. We identified the favorable outcome (drawing a green token) and the total possible outcomes (drawing any token). We then calculated the ratio of favorable outcomes to total outcomes, which gave us the probability of 8/30. Finally, we simplified this fraction to 4/15. This answer highlights the importance of careful calculation and simplification in probability problems. Choosing the correct answer requires not only understanding the concept of probability but also being able to perform accurate calculations. The distractors, A. 4/11 and C., are designed to mislead students who may make common errors in calculating probabilities. For example, 4/11 might be obtained if the student incorrectly considered the number of non-green tokens in the denominator. Understanding why the other options are incorrect is just as important as understanding why the correct answer is correct. This helps to solidify the student's understanding of the underlying concepts and prevent them from making similar errors in the future. The process of arriving at the correct answer reinforces the importance of a systematic approach to problem-solving. By breaking down the problem into smaller steps, such as identifying the favorable outcomes, identifying the total possible outcomes, and calculating the ratio, we can solve even complex probability problems with confidence. This step-by-step approach not only leads to the correct answer but also promotes a deeper understanding of the concepts involved. Therefore, the correct answer, 4/15, is not just a number; it's the result of a logical and systematic process of applying the principles of probability to the given problem. Understanding this process is the key to mastering probability and succeeding in similar problems.

H2: Common Mistakes to Avoid

When tackling probability problems, there are several common mistakes that students often make. Understanding these pitfalls can help you avoid them and improve your accuracy. One frequent error is failing to correctly identify the total number of possible outcomes. In our token-drawing problem, it's crucial to add up all the tokens of each color (10 blue + 8 green + 12 red = 30) to get the total number of tokens. Mistaking the total can lead to a completely incorrect probability calculation. Another common mistake is forgetting to simplify the fraction. While 8/30 is technically correct, it's best practice to simplify it to 4/15 for clarity and to match the answer choices. Failing to simplify can also make it harder to compare probabilities or use the result in further calculations. A third error is confusing the favorable outcome with something else. In this case, the favorable outcome is drawing a green token. Students might incorrectly focus on the number of non-green tokens or some other irrelevant quantity. It's essential to clearly define the event you're trying to find the probability for. Finally, students sometimes struggle with conditional probability, which involves the probability of an event happening given that another event has already occurred. While our problem doesn't explicitly involve conditional probability in the first draw, it's important to be aware of this concept for more complex problems. For example, the probability of drawing a second green token will change depending on whether a green token was drawn on the first draw. By being aware of these common mistakes and actively working to avoid them, you can significantly improve your accuracy in solving probability problems. Practice, careful attention to detail, and a clear understanding of the basic principles of probability are the keys to success.

H2: Extending the Problem: What if Marcus Draws Another Token?

Let's extend this token-drawing problem to make it even more interesting. Suppose Marcus draws a green token on his first draw, and he doesn't replace it. Now, what is the probability that he will draw another green token on his second draw? This introduces the concept of conditional probability, where the probability of an event depends on the outcome of a previous event. After Marcus draws a green token and doesn't replace it, the total number of tokens in the bag decreases by one, from 30 to 29. Crucially, the number of green tokens also decreases by one, from 8 to 7. This change in the composition of the bag directly affects the probability of drawing another green token. The probability of drawing a second green token is now the ratio of the new number of green tokens (7) to the new total number of tokens (29). Therefore, the probability is 7/29. This calculation demonstrates the importance of considering the changes in the sample space when dealing with conditional probability. The sample space is the set of all possible outcomes, and in this case, it changes after the first draw. Extending the problem in this way allows us to explore more complex probability scenarios and develop a deeper understanding of how events can influence each other. We can further extend the problem by asking about the probability of drawing a specific sequence of colors, such as green then red, or blue then green then blue. These types of problems require us to multiply probabilities together, taking into account the changing sample space after each draw. By exploring these extensions, we can solidify our understanding of probability and develop problem-solving skills that can be applied to a wide range of situations. This extended problem provides a valuable exercise in applying the principles of conditional probability and highlights the dynamic nature of probability calculations.

H2: Conclusion: Mastering Probability Through Practice

In conclusion, understanding probability is essential for success in mathematics and various real-world applications. This token-drawing problem provides a clear and accessible example of how to calculate probabilities, considering both the initial conditions and the impact of previous events. By carefully defining the problem, identifying favorable and total outcomes, and simplifying fractions, we can accurately determine the likelihood of specific events. The correct answer to the initial question, the probability of Marcus drawing a green token on the first draw, is 4/15. However, the true value lies not just in finding the answer but in understanding the process. Avoiding common mistakes, such as miscalculating the total number of outcomes or failing to simplify fractions, is crucial for accuracy. Extending the problem to include conditional probability, such as calculating the probability of drawing a second green token after the first one is removed, further deepens our understanding. The key to mastering probability is practice. By working through various problems, you can develop your problem-solving skills and build confidence in your ability to apply probability concepts. From simple scenarios like token draws to more complex situations involving multiple events and conditional probabilities, a solid foundation in probability will serve you well. Remember to always break down the problem into smaller steps, clearly identify the favorable and total outcomes, and consider how previous events might influence subsequent probabilities. With consistent practice and a clear understanding of the underlying principles, you can confidently tackle any probability problem that comes your way. So, keep practicing, keep exploring, and keep mastering the fascinating world of probability!