Marble Math How Many Red Marbles Needed For A 3/5 Probability

Let's embark on a mathematical journey into the world of probability, where we'll tackle a classic problem involving marbles and probabilities. This question delves into the core concepts of probability, requiring us to understand how adding elements to a set affects the likelihood of selecting a specific element. By dissecting this problem, we'll not only arrive at the correct answer but also gain a deeper appreciation for the principles that govern probability.

Problem Statement

Our problem presents a scenario involving a bag filled with marbles of different colors. Specifically, we have:

• 8 red marbles
• 9 yellow marbles
• 7 green marbles

This gives us a total of 24 marbles (8 + 9 + 7) initially in the bag. The central question is: How many additional red marbles must we add to the bag so that the probability of randomly drawing a red marble becomes $rac{3}{5}$? This seemingly simple question requires a careful application of probability principles and algebraic manipulation.

Understanding Probability

Before diving into the solution, let's refresh our understanding of probability. In simple terms, probability is the measure of the likelihood of an event occurring. It is expressed as a ratio, where the numerator represents the number of favorable outcomes (outcomes where the event occurs), and the denominator represents the total number of possible outcomes. Mathematically, the probability of an event ${ E }$ is given by:

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

In our marble problem, the event we're interested in is drawing a red marble. The favorable outcomes are the red marbles in the bag, and the total possible outcomes are all the marbles in the bag. Our goal is to find the number of additional red marbles that will make the probability of drawing a red marble equal to $rac{3}{5}$.

Setting up the Equation

Let's denote the number of additional red marbles we need to add as ${ x }$. After adding these ${ x }$ marbles, the total number of red marbles in the bag will be ${ 8 + x }$. The total number of marbles in the bag will also increase to ${ 24 + x }$. According to the problem, we want the probability of drawing a red marble to be $rac{3}{5}$. Therefore, we can set up the following equation:

$$\frac{8 + x}{24 + x} = \frac{3}{5}$$

This equation forms the crux of our solution. It mathematically represents the condition we want to satisfy: the probability of drawing a red marble after adding ${ x }$ red marbles should be equal to $rac{3}{5}$. Now, our task is to solve this equation for ${ x }$.

Solving for x

To solve the equation, we'll use cross-multiplication, a fundamental technique for handling proportions. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal. Applying this to our equation, we get:

$$5(8 + x) = 3(24 + x)$$

Next, we distribute the numbers on both sides of the equation:

$$40 + 5x = 72 + 3x$$

Now, we want to isolate ${ x }$ on one side of the equation. We can do this by subtracting ${ 3x }$ from both sides:

$$40 + 2x = 72$$

Next, subtract 40 from both sides:

$$2x = 32$$

Finally, divide both sides by 2 to solve for ${ x }$:

$$x = 16$$

Thus, we find that we need to add 16 additional red marbles to the bag to achieve a probability of $rac{3}{5}$ for drawing a red marble.

Verifying the Solution

It's always a good practice to verify our solution to ensure it makes sense in the context of the problem. If we add 16 red marbles to the bag, we'll have:

• 8 + 16 = 24 red marbles
• 24 + 16 = 40 total marbles

The probability of drawing a red marble is then:

$$\frac{24}{40} = \frac{3}{5}$$

This confirms that our solution, ${ x = 16 }$, is correct. Adding 16 red marbles indeed results in a probability of $rac{3}{5}$ for drawing a red marble.

Conclusion

In this problem, we've successfully navigated the realm of probability, determining the number of additional red marbles needed to achieve a specific probability outcome. The solution involved setting up an equation based on the fundamental principles of probability and then solving it using algebraic techniques. The answer, 16 red marbles, not only satisfies the mathematical condition but also provides a tangible understanding of how changing the composition of a set affects probabilities.

This exercise highlights the importance of understanding probability in various real-world scenarios, from games of chance to statistical analysis. By mastering these concepts, we equip ourselves with the tools to make informed decisions and predictions in a world governed by uncertainty.

Delving deeper into probability, let's explore some key concepts that are essential for understanding and solving problems like the marble scenario we just tackled. These concepts form the bedrock of probability theory and are widely applicable in various fields, including statistics, finance, and decision-making.

1. Sample Space and Events

The sample space is the set of all possible outcomes of an experiment. In our marble problem, the sample space is the set of all possible marbles we could draw from the bag. Initially, this included 8 red, 9 yellow, and 7 green marbles. After adding more red marbles, the sample space expanded to include the additional marbles.

An event is a subset of the sample space, representing a specific outcome or a set of outcomes we're interested in. In our case, the event of interest was drawing a red marble. The probability of an event is the ratio of the number of outcomes in the event to the total number of outcomes in the sample space.

2. Basic Probability Formula

As we discussed earlier, the basic formula for calculating the probability of an event ${ E }$ is:

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

This formula is the cornerstone of probability calculations. It provides a straightforward way to determine the likelihood of an event if we know the number of favorable outcomes and the total number of possible outcomes.

3. Independent and Dependent Events

Events can be classified as either independent or dependent. Independent events are those where the outcome of one event does not affect the outcome of another event. For example, flipping a coin multiple times are independent events, as the result of one flip does not influence the result of subsequent flips.

Dependent events, on the other hand, are those where the outcome of one event does affect the outcome of another event. Drawing marbles from a bag without replacement is a classic example of dependent events. If we draw a red marble and don't put it back, the probability of drawing another red marble on the next draw changes.

4. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as ${ P(A|B) }$, which reads as