Probability Of Drawing A Yellow And Red Marble An Explanation

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In the realm of probability, understanding how to calculate the likelihood of specific events is crucial. This article delves into a classic probability problem involving marbles of different colors in a bag. We will explore the steps to determine the probability of drawing one yellow and one red marble when two marbles are chosen at random. Understanding these concepts is foundational for various applications, from statistical analysis to decision-making in everyday situations. Let’s break down the problem, identify the key elements, and construct an expression that accurately represents the probability we seek. By the end of this exploration, you'll have a clear understanding of how to approach similar probability questions and a solid grasp of the underlying principles involved.

Our specific probability challenge involves a bag filled with an assortment of marbles: eight are yellow, nine are green, three are purple, and five are red. The question at hand is: if we randomly select two marbles from the bag, what expression represents the probability that one of these marbles is yellow and the other is red? This is a classic scenario in probability theory, requiring us to consider combinations and the fundamental principles of calculating probabilities. We will navigate through the process of identifying the total possible outcomes and the specific outcomes that meet our criteria, ultimately constructing an expression that precisely answers the question.

Before diving into the specifics of our marble problem, let's lay a solid foundation by revisiting the fundamentals of probability. Probability, at its core, is a numerical measure that expresses the likelihood of a particular event occurring. It's quantified as a value between 0 and 1, inclusive. A probability of 0 signifies impossibility – the event will not occur under any circumstances. Conversely, a probability of 1 indicates certainty – the event is guaranteed to happen. Values in between represent varying degrees of likelihood; for example, a probability of 0.5 (or 50%) suggests an equal chance of the event occurring or not occurring.

The general formula for calculating probability is elegantly straightforward:

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

Here's a breakdown of the key components:

  • Number of favorable outcomes: This is the count of outcomes that align with the specific event we are interested in. For instance, in our marble problem, these would be the outcomes where we draw one yellow and one red marble.
  • Total number of possible outcomes: This is the comprehensive count of all possible outcomes that could arise from the scenario. In the marble example, this represents all possible pairs of marbles we could draw from the bag.

To illustrate this concept, consider a simple example: rolling a fair six-sided die. What's the probability of rolling a 4? There's only one favorable outcome (rolling a 4), and there are six total possible outcomes (rolling a 1, 2, 3, 4, 5, or 6). Thus, the probability is 1/6.

Understanding this fundamental formula and its components is crucial for tackling more complex probability problems, including the marble challenge we are about to unravel. By carefully identifying favorable and total outcomes, we can accurately calculate the probabilities of various events.

Before we can calculate the probability of drawing a yellow and a red marble, we need to determine the total number of marbles in the bag. This is a straightforward addition problem. We have:

  • Eight yellow marbles
  • Nine green marbles
  • Three purple marbles
  • Five red marbles

To find the total, we simply add these quantities together:

8+9+3+5=258 + 9 + 3 + 5 = 25

Therefore, there are a total of 25 marbles in the bag. This total number is a crucial piece of information as it forms the basis for calculating the total possible outcomes when we draw two marbles. Knowing the total number of marbles allows us to accurately determine the denominator in our probability calculation, which represents the entire sample space of possible outcomes.

Now that we know there are 25 marbles in the bag, the next step is to calculate the total number of ways we can choose two marbles. This is a classic combination problem, as the order in which we select the marbles does not matter. We're simply interested in the pair of marbles we end up with.

The formula for combinations is given by:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Where:

  • n is the total number of items (in our case, 25 marbles)
  • k is the number of items we are choosing (in our case, 2 marbles)
  • ! denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1)

Applying this formula to our problem, we want to find the number of ways to choose 2 marbles out of 25, which is written as $\binom{25}{2}$. Let's calculate this:

(252)=25!2!(25−2)!=25!2!23!=25×242×1=25×12=300\binom{25}{2} = \frac{25!}{2!(25-2)!} = \frac{25!}{2!23!} = \frac{25 \times 24}{2 \times 1} = 25 \times 12 = 300

Therefore, there are 300 different ways to choose two marbles from the bag. This number represents the total possible outcomes and will be the denominator in our probability calculation.

Our next key task is to determine the number of favorable outcomes, specifically, the scenarios where we draw one yellow marble and one red marble. To calculate this, we need to consider the number of ways we can choose one yellow marble from the available yellow marbles and one red marble from the available red marbles. This involves another application of combinations, but in a slightly different way.

We have:

  • Eight yellow marbles
  • Five red marbles

We want to choose one marble from each color. The number of ways to choose one yellow marble from eight is $\binom{8}{1}$, and the number of ways to choose one red marble from five is $\binom{5}{1}$.

Using the combination formula:

(81)=8!1!(8−1)!=8!1!7!=8\binom{8}{1} = \frac{8!}{1!(8-1)!} = \frac{8!}{1!7!} = 8

(51)=5!1!(5−1)!=5!1!4!=5\binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{5!}{1!4!} = 5

Now, to find the total number of ways to choose one yellow and one red marble, we multiply these two results together. This is because for each way we can choose a yellow marble, there are multiple ways to choose a red marble, and we need to account for all these combinations.

8×5=408 \times 5 = 40

Therefore, there are 40 favorable outcomes – 40 different ways to choose one yellow and one red marble from the bag. This number is the numerator in our probability calculation, representing the specific outcomes we are interested in.

Now that we have all the necessary components, we can construct the expression for the probability of drawing one yellow and one red marble. We recall the basic probability formula:

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

We have determined that:

  • The number of favorable outcomes (one yellow and one red marble) is 40.
  • The total number of possible outcomes (choosing any two marbles) is 300.

Therefore, the probability of drawing one yellow and one red marble is:

P(Y and R)=40300P(Y \text{ and } R) = \frac{40}{300}

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 20:

P(Y and R)=40÷20300÷20=215P(Y \text{ and } R) = \frac{40 \div 20}{300 \div 20} = \frac{2}{15}

Thus, the probability of drawing one yellow and one red marble is $\frac{2}{15}$. The expression that gives this probability is $\frac{40}{300}$, which simplifies to $\frac{2}{15}$.

In this exploration, we've successfully navigated a classic probability problem involving marbles of different colors in a bag. We meticulously calculated the probability of drawing one yellow and one red marble when two marbles are chosen at random. Our journey involved understanding the fundamentals of probability, determining the total number of marbles, calculating the total number of possible outcomes using combinations, and identifying the number of favorable outcomes that meet our specific criteria. Through these steps, we constructed the probability expression $\frac{40}{300}$, which simplifies to $\frac{2}{15}$.

This problem serves as an excellent illustration of how probability concepts can be applied to real-world scenarios. By breaking down complex problems into manageable steps and applying the appropriate formulas, we can effectively calculate the likelihood of specific events. The principles we've covered here are not only applicable to marble problems but also extend to a wide range of probability challenges, from card games to statistical analysis in various fields. Understanding these concepts empowers us to make informed decisions and predictions in situations involving uncertainty. Probability is a powerful tool, and mastering its fundamentals opens doors to deeper insights into the world around us.

Probability, Marbles, Yellow, Red, Combinations, Favorable Outcomes, Total Outcomes, Probability Expression, Calculation, Mathematics