Probability Of Drawing White Or Black Balls

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Hey guys! Let's dive into a classic probability problem. We're gonna figure out the odds of snagging some white or black balls from a mixed bag. This stuff is super useful, whether you're into statistics, trying to win a game, or just curious about how chance works. So, let's break down this problem step by step to make sure we understand it.

The Setup: Understanding the Ball Pit

Alright, imagine we have a big urn (like a fancy container) filled with colorful balls. Here's the breakdown of what's inside:

  • 9 White Balls: These are the ones we're hoping to grab (or at least some of them).
  • 9 Black Balls: Also, good to us! These will satisfy our goal of getting white or black.
  • 9 Blue Balls: Just chilling in there, not part of our immediate win condition.
  • 9 Red Balls: Same as the blue ones; they don't help us in this particular scenario.

So, in total, we've got 36 balls (9+9+9+9). The key thing here is that each ball is equally likely to be picked. We're picking balls randomly, meaning there's no trickery involved – it's all about pure chance! The core of probability problems lies in understanding the composition of the whole. The more balls of a certain color, the greater the likelihood of picking one of them. The probability is about what will happen, but we can't tell for certain what will happen.

The Question: Our Objective

Now, the question is: We randomly pull out 3 balls from the urn. What's the probability that ALL three balls we pick are either white OR black? We are not interested in the other colors. This is where we will use our heads and figure out the odds. This is what we need to figure out to solve our problem. The question could be phrased differently, but in essence, we are calculating the chances of an event happening. This means we are figuring out the likelihood of picking the right balls. Let's think through this! To calculate the odds, we're not just looking at the number of white and black balls; we have to account for the total number of ways we can pick any 3 balls out of 36. This is essential for calculating the correct probability.

Calculating the Possibilities: All Possible Combinations

Alright, let's start with the total number of ways we can pick 3 balls from 36, regardless of their color. This is a combination problem because the order in which we pick the balls doesn't matter. It is a mathematical calculation that tells us the total number of ways. We use something called the combination formula. The formula is:

C(n, k) = n! / (k!(n-k)!)

Where:

  • n is the total number of items (in our case, 36 balls).
  • k is the number of items we're choosing (in our case, 3 balls).
  • ! means factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).

So, plugging in our numbers, we get:

C(36, 3) = 36! / (3!(36-3)!)

Let's calculate this:

C(36, 3) = 36! / (3! * 33!)

C(36, 3) = (36 * 35 * 34) / (3 * 2 * 1)

C(36, 3) = 7140

So there are 7,140 different ways to pick 3 balls from the urn.

Success Cases: White and Black Combinations

Now, let's find out how many of those 7,140 combinations are successful. To do this, we need to think about how many ways we can pick 3 balls that are ALL white or black. Since we want either white or black, we can treat them as a single group. Together, there are 18 balls that meet our criteria (9 white + 9 black). We need to figure out the different ways we can choose 3 balls from these 18. Using the combination formula again:

C(18, 3) = 18! / (3!(18-3)!)

C(18, 3) = 18! / (3! * 15!)

C(18, 3) = (18 * 17 * 16) / (3 * 2 * 1)

C(18, 3) = 816

This means there are 816 successful combinations – ways to pick 3 balls that are either all white, all black, or a mix of white and black.

Putting it Together: Calculating the Probability

Finally, we can find the probability by dividing the number of successful outcomes (picking 3 white or black balls) by the total number of possible outcomes (picking any 3 balls). The probability is then calculated with the formula:

Probability = (Successful Outcomes) / (Total Possible Outcomes)

So, plugging in our numbers:

Probability = 816 / 7140

Probability ≈ 0.1143

This means there is approximately a 11.43% chance that all 3 balls you pick will be white or black. Isn't that interesting? Probability helps us with a range of things. Probability helps us predict what might happen in the future, based on past experiences. It is an amazing and useful tool!

Conclusion: Wrapping It Up

So, there you have it, guys! We have successfully figured out the probability of drawing white or black balls from our urn. We went through each step carefully, from understanding the situation to calculating the final answer. Probability is super interesting, and these kinds of problems help us see it in action. Remember that the more you practice, the easier these problems become. Keep at it, and you'll be acing these questions in no time!

Key Takeaways:

  • Combinations are key: Understanding how to calculate combinations is essential for probability problems.
  • Break it down: Break down the problem into smaller, manageable steps.
  • Practice makes perfect: The more you practice, the better you'll get at solving these types of problems.

Keep exploring, and happy calculating!