Probability Of Multiples Of 5 And 6 In 100 Balls

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Hey guys! Ever wondered about the chances of picking a specific number from a set? Let's dive into a super common probability question. This time, it involves figuring out the likelihood of selecting a ball that’s a multiple of both 5 and 6 from a collection of 100 balls. Sounds interesting? Let’s break it down!

Understanding the Problem

So, here’s the deal. Imagine you have a hundred balls, each neatly labeled from 1 to 100. Now, what are the odds that if you randomly grab one, the number on that ball is a multiple of both 5 and 6? Understanding this involves a bit of number theory and probability, but don't worry, it's simpler than it sounds. First off, when we say a number is a 'multiple of both 5 and 6', we're essentially looking for numbers that are divisible by both. Think of it like finding a common ground between two sets of numbers. Those divisible by 5 and also by 6. The key here is recognizing that a number divisible by both 5 and 6 must be divisible by their least common multiple (LCM). This is because the LCM is the smallest number that both 5 and 6 can divide into without leaving a remainder. We use the least common multiple, or LCM, to solve this problem. Understanding the LCM is crucial for solving this problem. Once you understand this, the problem becomes much easier.

Finding the Least Common Multiple (LCM)

Before we calculate probabilities, we need to find the LCM of 5 and 6. Here’s how you do it:

  1. Prime factorization of 5: 5 (since 5 is a prime number).
  2. Prime factorization of 6: 2 x 3.
  3. LCM is found by taking the highest power of each prime factor: 2 x 3 x 5 = 30.

So, the LCM of 5 and 6 is 30. This means we are looking for numbers that are multiples of 30.

Calculating the Probability

Alright, now that we know we're looking for multiples of 30, let's figure out how many of those exist within our set of 100 balls. Identifying these multiples is straightforward:

  1. List the multiples of 30 that are less than or equal to 100: 30, 60, 90.

So, we have three numbers (30, 60, and 90) that fit our criterion. Now, let's bring in the probability formula. Probability is defined as:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In our case:

  • Number of favorable outcomes (multiples of 30) = 3
  • Total number of possible outcomes (total number of balls) = 100

Plugging these values into the formula, we get:

Probability = 3 / 100 = 0.03

Therefore, the probability of drawing a ball with a number that is a multiple of both 5 and 6 is 0.03. This is a crucial step in solving the problem. Make sure you understand each step before moving on.

Why This Matters

You might be wondering, “Why bother with this?” Well, understanding probability and number theory is super useful in many real-world situations. From calculating risks in business to understanding statistics in sports, these concepts pop up everywhere. Plus, it's a great exercise for your brain!

Common Mistakes to Avoid

When tackling problems like these, it's easy to slip up. Here are a few common mistakes to watch out for:

  • Forgetting to find the LCM: Some people might try to find multiples of 5 and 6 separately and then combine them. This can lead to double-counting and incorrect results.
  • Misunderstanding the question: Always make sure you fully understand what the question is asking. In this case, it's crucial to recognize that the number must be a multiple of both 5 and 6.
  • Math errors: Simple arithmetic errors can throw off your entire calculation. Double-check your work, especially when dividing or multiplying.
  • Getting LCM wrong: Always double check the LCM of numbers. A wrong LCM will yield to the wrong answer.

Practical Applications of Probability

Okay, so we've solved the problem and avoided some common pitfalls. But how does this stuff actually apply in the real world? Here are a few examples:

  • Finance: In finance, understanding probability is crucial for assessing risk. For example, investors use probability to estimate the likelihood of different investment outcomes.
  • Insurance: Insurance companies rely heavily on probability to calculate premiums. They assess the likelihood of various events (like accidents or illnesses) and set premiums accordingly.
  • Games of chance: Of course, probability is fundamental to games of chance like poker or lotteries. Understanding the odds can help you make more informed decisions (though it won't guarantee a win!).
  • Science and Engineering: In science and engineering, probability is used in experimental design, data analysis, and quality control.

Practice Problems

Want to put your newfound knowledge to the test? Try these practice problems:

  1. What is the probability of drawing a ball with a number that is a multiple of both 4 and 6 from a set of 100 balls numbered 1 to 100?
  2. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a red marble?
  3. What is the probability of rolling an even number on a six-sided die?

Conclusion

So, there you have it! Calculating the probability of drawing a ball with a number that is a multiple of both 5 and 6 involves understanding LCM, identifying favorable outcomes, and applying the probability formula. Remember to avoid common mistakes and practice regularly to sharpen your skills. Probability problems might seem daunting at first, but with a bit of practice and the right approach, you'll be solving them like a pro in no time! Keep practicing, and you'll master these concepts. Understanding probability is a useful skill. Hope this helps, and happy calculating!