Probability Of Playing Only Volleyball: Step-by-Step Solution

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Hey guys! Let's break down this probability problem step-by-step. We've got a group of students who play volleyball, soccer, or both, and we need to figure out the chance of randomly picking someone who only plays volleyball. It might sound tricky at first, but don't worry, we'll make it super clear.

Understanding the Problem

Okay, so here's the deal. In this problem, we need to calculate a probability. Remember, probability is just the chance of something happening. It’s often expressed as a percentage, and we find it by dividing the number of successful outcomes by the total number of possible outcomes. In our case, the successful outcome is picking a student who only plays volleyball, and the total number of possible outcomes is the total number of students.

We know a few key things:

  • There are 80 students in total. This is our total number of possible outcomes.
  • 20 students play both soccer and volleyball.
  • 30 students play only volleyball. This is our number of successful outcomes.

The question asks: If we pick a student at random, what’s the probability they play only volleyball? We have three possible answers: 25%, 30%, or 37.5%. To solve this, we will use the basic probability formula: $Probability = \frac{Number of Successful Outcomes}{Total Number of Possible Outcomes}$

Calculating the Probability

Now that we understand the problem, let’s crunch the numbers. This is where it gets satisfying!

Step 1: Identify the Numbers We Need

We already figured this out, but let’s recap:

  • Number of students who play only volleyball (successful outcomes): 30
  • Total number of students (possible outcomes): 80

Step 2: Apply the Probability Formula

Now we plug these numbers into our probability formula:

Probability=3080Probability = \frac{30}{80}

Step 3: Simplify the Fraction

To make things easier, we can simplify the fraction. Both 30 and 80 are divisible by 10, so let’s divide both by 10:

Probability=38Probability = \frac{3}{8}

Step 4: Convert to Percentage

To compare our answer to the options (25%, 30%, 37.5%), we need to convert the fraction to a percentage. To do this, we divide the numerator (3) by the denominator (8) and then multiply by 100:

Probability=(3Ă·8)Ă—100Probability = (3 Ă· 8) Ă— 100

Probability=0.375Ă—100Probability = 0.375 Ă— 100

Probability=37.5Probability = 37.5

So, the probability of randomly picking a student who only plays volleyball is 37.5%.

Step 5: Choose the Correct Answer

Looking back at our options, we see that C) 37.5% is the correct answer!

Why This Matters: Real-World Probability

Understanding probability isn't just about solving math problems; it's something we use all the time in real life, even if we don't realize it. Think about it: when you're deciding whether to carry an umbrella, you're considering the probability of rain. When you're playing a game, you're thinking about the probability of rolling a certain number on a dice or drawing a specific card.

In this problem, we applied a basic probability concept to a scenario involving students and sports. But the same principles can be used in tons of different situations, from business decisions to scientific research. For instance, businesses use probability to estimate the chances of a product being successful, and scientists use it to analyze the results of experiments.

Knowing how to calculate and interpret probabilities helps you make informed decisions and understand the world around you. It's a valuable skill, whether you're planning a picnic, investing money, or just trying to figure out the best route to work. So, keep practicing these types of problems, and you'll become a probability pro in no time!

Common Mistakes and How to Avoid Them

Probability problems can be tricky, and it’s easy to make a mistake if you're not careful. Here are a few common pitfalls and how to avoid them:

  1. Confusing “Only” with “Total”: In this problem, it’s crucial to distinguish between the number of students who play only volleyball (30) and the total number of students who play volleyball (which would include those who play both). Make sure you're focusing on the specific group the question asks about. Read the question carefully, paying attention to words like “only,” “and,” and “or.”

  2. Using the Wrong Denominator: The denominator in the probability fraction should always be the total number of possible outcomes. In this case, it’s the total number of students (80). Don’t be tempted to use a smaller number, like the number of students who play volleyball, as your denominator. Always identify the overall group you're selecting from.

  3. Forgetting to Simplify or Convert: Sometimes, the answer you get as a fraction isn’t directly comparable to the options. Remember to simplify your fraction and convert it to a percentage or decimal if necessary. This makes it easier to match your answer to the choices provided. Practice simplifying fractions and converting them to percentages regularly.

  4. Misinterpreting the Question: Probability questions can sometimes be worded in a way that’s a bit confusing. Take your time to read the question carefully and make sure you understand what it’s asking before you start calculating. Break the problem down into smaller parts. What information do you have? What are you trying to find?

  5. Not Checking Your Answer: Once you've calculated your answer, take a moment to see if it makes sense in the context of the problem. Does the probability you calculated seem reasonable? If it’s a very high or very low probability, ask yourself if that aligns with the situation described. A quick check can help you catch simple errors.

By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering probability problems!

Alternative Approaches

While we solved this problem using the direct probability formula, there are other ways you could approach it. Understanding different methods can give you a deeper grasp of the concept and help you tackle similar problems more confidently.

  1. Visual Representation (Venn Diagram)

A Venn diagram can be a super helpful way to visualize the information in this type of problem. You can draw two overlapping circles, one representing students who play soccer and the other representing students who play volleyball. The overlapping section represents students who play both.

  • Fill in the overlapping section with the number of students who play both (20).
  • Fill in the “only volleyball” section with the number of students who play only volleyball (30).
  • You could also calculate the number of students who play only soccer if the problem provided that information.

The total number of students (80) would be the sum of all sections in the Venn diagram, plus any students who don’t play either sport. By visually organizing the information, you can easily see the number of successful outcomes (30) and the total number of outcomes (80), making the probability calculation straightforward.

  1. Thinking in Proportions

Instead of directly using the probability formula, you can think about the problem in terms of proportions. We know that 30 out of 80 students play only volleyball. This means the proportion of students who play only volleyball is 30/80. To find the probability as a percentage, we simply need to express this proportion as a percentage, which we already did in our main solution.

This approach highlights that probability is essentially a way of expressing proportions. It can be particularly useful when dealing with problems where you’re given ratios or proportions directly.

  1. Step-by-Step Reasoning

You can also break down the problem into a series of logical steps without relying heavily on formulas.

  • “We want to find the chance of picking someone who only plays volleyball.”
  • “There are 30 people who fit this description.”
  • “There are 80 people in total we could pick.”
  • “So, the chance is 30 out of 80.”
  • “We can write this as a fraction: 30/80.”
  • “Now, let’s simplify and convert to a percentage.”

This method emphasizes the reasoning behind the calculation and can be helpful if you struggle to remember formulas. It encourages you to think through the problem logically, step by step.

By exploring these alternative approaches, you'll not only reinforce your understanding of probability but also develop problem-solving skills that can be applied to a wide range of situations. Each method offers a slightly different perspective, and finding the one that clicks best with your thinking style can make tackling probability problems much easier.

Let's Wrap It Up

So, there you have it! We've successfully calculated the probability of randomly selecting a student who only plays volleyball. Remember, the key is to understand the problem, identify the relevant numbers, and apply the correct formula. Don't be afraid to try different methods and practice regularly. You've got this!

Probability is a fundamental concept that pops up in all sorts of places, from games of chance to real-world decision-making. By mastering these basics, you're setting yourself up for success in many areas. Keep practicing, keep asking questions, and you'll become a probability whiz in no time. Good luck, and happy calculating!