Calculating Marble Probability Determining The Expression For Red And Blue Marble Draws

When tackling probability problems, it's crucial to understand the underlying principles and apply them systematically. In this article, we will explore a classic probability scenario involving marbles of different colors in a bag. We will dissect the problem, identify the key steps, and provide a detailed explanation of the solution. Our focus will be on calculating the probability of drawing one red marble and one blue marble from a bag containing a mix of colored marbles. This involves understanding combinations, permutations, and the fundamental principles of probability. Let's embark on this mathematical journey to unravel the complexities of probability calculations.

Problem Statement

Imagine a bag filled with an assortment of marbles: 10 red marbles, 15 yellow marbles, 5 green marbles, and 20 blue marbles. Now, suppose we draw two marbles from this bag. Our mission is to determine the expression that accurately represents the probability of drawing one red marble and one blue marble. This seemingly simple scenario opens the door to a fascinating exploration of probability concepts and techniques. To solve this, we'll need to consider the total number of marbles, the number of ways to choose one red and one blue marble, and the total number of ways to choose any two marbles. This problem not only tests our understanding of probability but also our ability to apply combinatorial principles in a real-world context. Let's delve deeper into the solution and uncover the intricacies of probability calculations.

Breaking Down the Problem

To solve this probability problem effectively, let's break it down into manageable steps. First, we need to determine the total number of marbles in the bag. This will serve as the foundation for our probability calculations. We have 10 red marbles, 15 yellow marbles, 5 green marbles, and 20 blue marbles. Adding these up, we get a total of 50 marbles. Next, we need to calculate the number of ways to choose one red marble and one blue marble. This involves considering the number of red marbles (10) and the number of blue marbles (20). We can choose one red marble in 10 ways and one blue marble in 20 ways. Therefore, the number of ways to choose one red and one blue marble is 10 multiplied by 20, which equals 200. Finally, we need to calculate the total number of ways to choose any two marbles from the bag. This is a combination problem, as the order in which we draw the marbles doesn't matter. We can use the combination formula, which is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items we are choosing. In this case, n = 50 and r = 2. Plugging these values into the formula, we get 50C2 = 50! / (2! * 48!) = (50 * 49) / 2 = 1225. With these components in place, we can now formulate the expression for the probability of drawing one red marble and one blue marble.

Calculating the Probability

With the groundwork laid, let's calculate the probability of drawing one red marble and one blue marble. As we've established, there are 200 ways to choose one red and one blue marble. This is the number of favorable outcomes. The total number of ways to choose any two marbles from the bag is 1225. This represents the total possible outcomes. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of drawing one red marble and one blue marble is 200 / 1225. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 25. Simplifying the fraction, we get 8 / 49. This is the probability of drawing one red marble and one blue marble from the bag. However, the original question asks for the expression that represents this probability, not the simplified value. Therefore, we need to look for an expression that is equivalent to 200 / 1225. Let's examine the given options and see which one matches our calculated probability expression. The key is to understand how the numbers 200 and 1225 were derived and to match that logic in the given expressions.

Understanding Permutations and Combinations

Before we delve into the expressions, let's clarify the concepts of permutations and combinations, as they are fundamental to probability calculations. A permutation is an arrangement of objects in a specific order. The order matters in permutations. For example, if we have three letters, A, B, and C, the permutations of choosing two letters are AB, BA, AC, CA, BC, and CB. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items we are choosing. On the other hand, a combination is a selection of objects where the order does not matter. In our marble problem, the order in which we draw the marbles doesn't matter; we are only concerned with the colors of the marbles we draw. The formula for combinations is nCr = n! / (r! * (n-r)!). In our case, we used the combination formula to calculate the total number of ways to choose two marbles from the bag. Understanding the distinction between permutations and combinations is crucial for solving probability problems accurately. Now that we have a solid grasp of these concepts, let's return to the expressions and determine which one represents the probability of drawing one red marble and one blue marble.

Analyzing the Expressions

Now, let's analyze the given expression $\frac{30 P_2}{50 P_2}$ and determine if it correctly represents the probability of drawing one red and one blue marble. The expression uses the notation "P_2", which likely refers to a permutation of 2 items. Let's break down the expression step by step. The numerator, 30 P_2, suggests a permutation calculation involving the number 30. However, in our problem, the numbers 10 (red marbles) and 20 (blue marbles) are the key quantities for calculating the favorable outcomes. The number 30 doesn't directly relate to the number of red or blue marbles, so this part of the expression raises a flag. The denominator, 50 P_2, suggests a permutation of 2 items chosen from a set of 50. This could potentially represent the total number of ways to choose two marbles from the bag, considering the total number of marbles is 50. However, we need to carefully examine whether permutation is the appropriate operation here, as the order of drawing the marbles does not matter. Remember, we calculated the total number of ways to choose two marbles using combinations (50C2), not permutations. To determine if the expression is correct, we need to calculate the value of 30 P_2 and 50 P_2 and see if their ratio matches our previously calculated probability of 200 / 1225. If the ratio doesn't match, then the expression is incorrect. Let's proceed with the calculations and compare the results.

Evaluating the Given Expression

Let's evaluate the given expression $\frac{30 P_2}{50 P_2}$ to see if it matches our calculated probability. First, we need to understand the permutation notation. The expression 30 P_2 represents the number of permutations of choosing 2 items from a set of 30. Using the permutation formula, nPr = n! / (n-r)!, we have 30 P_2 = 30! / (30-2)! = 30! / 28! = 30 * 29 = 870. Next, we evaluate 50 P_2, which represents the number of permutations of choosing 2 items from a set of 50. Using the same formula, we have 50 P_2 = 50! / (50-2)! = 50! / 48! = 50 * 49 = 2450. Now, we have the expression $\frac{870}{2450}$. Let's simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10. This gives us $\frac{87}{245}$. Comparing this fraction to our previously calculated probability of $\frac{200}{1225}$, which simplifies to $\frac{8}{49}$, we can see that they are not equal. Therefore, the expression $\frac{30 P_2}{50 P_2}$ does not correctly represent the probability of drawing one red marble and one blue marble. This highlights the importance of carefully analyzing the given expressions and verifying them against our calculated results. The use of permutations in this expression, when combinations are more appropriate, is a key reason why it doesn't match the correct probability. Let's recap the correct approach to ensure we have a clear understanding of the solution.

Recapping the Solution

To recap, the correct approach to solving this probability problem involves the following steps: 1. Determine the total number of marbles: We have 10 red, 15 yellow, 5 green, and 20 blue marbles, totaling 50 marbles. 2. Calculate the number of ways to choose one red and one blue marble: This is 10 (ways to choose a red marble) * 20 (ways to choose a blue marble) = 200. 3. Calculate the total number of ways to choose any two marbles: This is a combination problem, so we use 50C2 = 50! / (2! * 48!) = (50 * 49) / 2 = 1225. 4. Calculate the probability: This is the number of favorable outcomes (200) divided by the total number of possible outcomes (1225), giving us 200 / 1225. 5. Simplify the fraction (optional): 200 / 1225 simplifies to 8 / 49. The expression $\frac{30 P_2}{50 P_2}$ was evaluated and found to be incorrect because it uses permutations instead of combinations and doesn't accurately reflect the number of red and blue marbles. The correct expression should represent the ratio of favorable outcomes (choosing one red and one blue marble) to the total possible outcomes (choosing any two marbles). This problem underscores the significance of understanding the fundamental principles of probability, including combinations and permutations, and applying them correctly to solve real-world scenarios. By breaking down the problem into manageable steps and carefully considering each aspect, we can arrive at the correct solution and gain a deeper understanding of probability calculations.

Conclusion

In conclusion, calculating probabilities involving combinations of items, such as marbles in a bag, requires a systematic approach. We must carefully identify the favorable outcomes and the total possible outcomes, and then apply the appropriate combinatorial principles. The expression $\frac{30 P_2}{50 P_2}$ does not accurately represent the probability of drawing one red marble and one blue marble in this scenario because it incorrectly uses permutations and doesn't align with the specific numbers of red and blue marbles. The correct approach involves calculating the number of ways to choose one red and one blue marble and dividing it by the total number of ways to choose any two marbles. This problem serves as a valuable exercise in understanding and applying probability concepts, highlighting the importance of precision and accuracy in mathematical calculations. By mastering these principles, we can confidently tackle a wide range of probability problems and gain a deeper appreciation for the role of probability in our daily lives.