Probability Of Drawing A Red And Blue Marble Expression

by Kenji Nakamura 56 views

Hey guys! Let's dive into this fascinating probability problem. Imagine a bag filled with colorful marbles – 10 red, 15 yellow, 5 green, and a whopping 20 blue ones. Now, we're going to reach in and grab two marbles. The big question is: what's the probability that one of those marbles will be red, and the other will be blue? Let's break it down step by step and make sure we nail this.

Understanding the Basics of Probability

Before we jump into the specifics, it’s super important to grasp the fundamentals of probability. Probability, at its core, is all about figuring out how likely something is to happen. We usually express it as a fraction, a decimal, or a percentage. In math terms, it's the ratio of the number of favorable outcomes to the total number of possible outcomes. Think of it like this: if there's only one way to win a game, and ten different things could happen, your chance of winning is 1 out of 10, or 1/10.

So, what are favorable outcomes? These are the specific results we're interested in. In our case, it's drawing one red marble and one blue marble. Total possible outcomes are all the different combinations of marbles we could draw. To really get this, let's think about simpler examples first. Suppose we only had red and blue marbles, and we were just picking one. The favorable outcome would be picking either a red or a blue, and the total outcomes would be the total number of marbles. But with two marbles, it gets a bit more complex. We have to consider combinations, and that’s where things like permutations and combinations come into play.

Probability isn't just some abstract math concept; it's all around us! From weather forecasts predicting the chance of rain to the odds of winning the lottery, it plays a role in many aspects of our lives. Understanding these basic principles will not only help us solve this marble problem but also give us a powerful tool for analyzing and making sense of the world around us. Probability helps in making informed decisions. For instance, when assessing risk in investments or planning an event, understanding the likelihood of different scenarios is invaluable. This foundational understanding of probability is your first step in tackling more complex problems, so let's keep building on it!

Calculating the Probability: Step-by-Step

Okay, let’s get down to the nitty-gritty and figure out this marble probability. We know we want to find the probability of drawing one red marble and one blue marble. But how do we actually calculate that? Well, we need to consider a couple of things. Firstly, there are two ways this can happen: we can either draw a red marble first and then a blue marble, or we can draw a blue marble first and then a red marble. These are two distinct scenarios, and we need to account for both of them.

First Scenario: Red then Blue

Let's start with the chance of drawing a red marble first. We have 10 red marbles out of a total of 50 marbles (10 red + 15 yellow + 5 green + 20 blue). So, the probability of picking a red marble first is 10/50. Now, let’s say we’ve successfully picked a red marble. We’ve taken one marble out of the bag, so there are only 49 marbles left. Crucially, the number of blue marbles hasn't changed – there are still 20 blue marbles. So, the probability of picking a blue marble next is 20/49.

To find the probability of both these events happening in sequence (red then blue), we multiply the individual probabilities: (10/50) * (20/49). This gives us the probability of the first scenario.

Second Scenario: Blue then Red

Now, let's think about the other possibility: drawing a blue marble first. There are 20 blue marbles out of 50 total, so the probability of picking a blue marble first is 20/50. If we've picked a blue marble, there are again 49 marbles left in the bag. This time, the number of red marbles hasn’t changed – there are still 10 red marbles. So, the probability of picking a red marble next is 10/49.

Just like before, to find the probability of both these events happening in sequence (blue then red), we multiply the individual probabilities: (20/50) * (10/49). This gives us the probability of the second scenario.

Combining the Scenarios

We've calculated the probability of each scenario separately. But remember, we want to know the probability of either one happening. To find the total probability, we add the probabilities of the two scenarios together: (10/50) * (20/49) + (20/50) * (10/49). This will give us the final probability of drawing one red marble and one blue marble, no matter the order. By breaking it down into these steps, we make sure we're accounting for every possibility and getting an accurate answer. This method is super handy for all sorts of probability problems, so remember this technique!

The Expression Representing the Probability

Alright, let’s piece together the expression that represents the probability we've just calculated. We figured out that there are two scenarios: drawing a red marble first and then a blue marble, or drawing a blue marble first and then a red marble. We also calculated the probabilities for each of these scenarios. Now, how do we write that all in one neat mathematical expression?

Remember, we calculated the probability of the first scenario (red then blue) as (10/50) * (20/49), and the probability of the second scenario (blue then red) as (20/50) * (10/49). Since we want the probability of either of these scenarios happening, we add them together. So, the expression looks like this:

(10/50) * (20/49) + (20/50) * (10/49)

Now, let's see if we can simplify this expression a bit. Notice that both terms have the same denominators, 50 and 49. Also, they both have the same numerators, just in a different order (10 * 20 and 20 * 10). This means we can rewrite the expression as:

2 * (10/50) * (20/49)

Why is this helpful? Well, it makes the expression a bit more compact and easier to understand. It also highlights the symmetry of the problem – it doesn't matter which color we draw first, the probability is the same.

But wait, there's another way to think about this! We can use the concept of combinations to solve this problem. A combination is a way of selecting items from a group where the order doesn't matter. In our case, we want to choose one red marble out of 10 and one blue marble out of 20. The total number of ways to choose two marbles out of 50 is given by the combination formula.

The number of ways to choose one red marble out of 10 is C(10, 1) = 10. The number of ways to choose one blue marble out of 20 is C(20, 1) = 20. The total number of ways to choose two marbles out of 50 is C(50, 2) = (50 * 49) / (2 * 1) = 1225. So, the probability can also be expressed as:

(10 * 20) / 1225

This might look different from our first expression, but it represents the same probability. It's just another way of looking at the problem, using the language of combinations. In the world of probability, there are often multiple paths to the same answer, and understanding these different approaches can give you a deeper understanding of the problem.

Comparing to the Given Options

So, we've arrived at our expression that represents the probability, and we've even seen a couple of different ways to write it. Now comes the crucial step: comparing our expression to the options provided in the original problem. This is where we put our mathematical detective hats on and see which option matches our result. The original question provided the following expression as a possible answer:

30P250P2\frac{30 P _2}{{ }_{50}P _2}

Let's break down what this expression means. The notation “P” usually refers to permutations. A permutation is a way of selecting items from a group where the order does matter. So, 30P2 means the number of ways to choose 2 items from a group of 30, where the order matters. Similarly, 50P2 means the number of ways to choose 2 items from a group of 50, where the order matters.

Let’s calculate these permutations: 30P2 = 30 * 29 = 870, and 50P2 = 50 * 49 = 2450. So, the given expression becomes 870 / 2450. Now, how does this compare to our expressions?

Our expressions were: 2 * (10/50) * (20/49) and (10 * 20) / 1225. If we calculate these, we get: 2 * (1/5) * (20/49) = 40/245 and 200 / 1225. If we simplify these fractions, we get 40/2450 = 4/245 and 200/1225 = 8/49. Now, let’s look at 870 / 2450. We can simplify this by dividing both numerator and denominator by 10, giving us 87/245. This is clearly different from our simplified probability of 4/245 or 8/49.

So, the expression $\frac{30 P 2}{{ }{50}P _2}$ doesn't match the probability we calculated. It's super important to go through this comparison step, because sometimes the options can look similar, but only one will be the exact match. In this case, we’ve shown that the provided option is not the correct representation of the probability of drawing one red marble and one blue marble.

Conclusion: Mastering Probability Problems

We did it! Guys, we've successfully navigated through this marble probability problem. We started with the basics of probability, broke down the problem into manageable scenarios, calculated the probabilities for each, and then combined them to find the overall probability. We even compared our result to a given option and saw why it didn't match. This journey highlights some key strategies for tackling probability problems.

Key Takeaways

  1. Understand the Fundamentals: Make sure you have a solid grasp of the basic principles of probability, including what favorable outcomes and total possible outcomes are. This foundation is crucial for tackling more complex problems.
  2. Break It Down: When faced with a complex probability problem, break it down into smaller, more manageable scenarios. This makes it easier to calculate the probabilities for each part and then combine them.
  3. Consider All Possibilities: Don't forget to consider all the different ways an event can occur. In our case, we had to think about drawing red then blue, and blue then red.
  4. Use the Right Tools: Depending on the problem, you might need to use combinations or permutations. Understand the difference between them and when to apply each one.
  5. Simplify and Compare: Once you've calculated the probability, simplify your expression and compare it to the given options. This is a crucial step to ensure you've arrived at the correct answer.

Probability can seem daunting at first, but with practice and a systematic approach, you can master these types of problems. Remember to take it step by step, break it down, and don't be afraid to think through all the possibilities. By using these strategies, you'll be able to confidently tackle any probability challenge that comes your way. So keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!

This was quite the marble adventure, wasn't it? Probability problems are like puzzles, and it's so satisfying when you finally fit all the pieces together. So, next time you see a probability question, remember these steps and dive in – you might just surprise yourself with what you can achieve. Keep up the awesome work!