Probability Of Selecting Tiles: A Step-by-Step Solution
Hey guys! Let's dive into this probability problem where Harold is selecting tiles. We need to figure out the chance that he picks a pink square tile and either a red or blue round tile. Sounds like fun, right? Probability can seem intimidating, but breaking it down step by step makes it super manageable. We'll walk through each part, making sure you understand the logic behind it. So, grab your thinking caps, and let's get started!
Understanding the Problem
First, let's make sure we're all on the same page. Harold has two sets of tiles: square tiles and round tiles. We know the number of each color of square tiles: 20 green, 16 pink, 9 red, and 10 yellow. We also have information about the round tiles (though the specifics aren't provided in the title, we'll assume we have that information when solving the problem). The key here is that Harold is making two selections: one square tile and one round tile. This means we're dealing with compound probability, where we need to consider the probability of two independent events happening.
When tackling any probability problem, it's always a good idea to break it down into smaller, more manageable steps. What are we trying to find? The probability of two things happening together: (1) selecting a pink square tile, and (2) selecting either a red or a blue round tile. The word "and" is super important here because it tells us we'll likely need to multiply probabilities at some point. Now, let’s think about how we calculate probability in general. It's simply the number of favorable outcomes divided by the total number of possible outcomes. So, for each part of Harold's selection, we need to figure out these two numbers: What are the chances of grabbing a pink square tile out of all the square tiles? And what are the chances of picking a red or blue round tile from the round tile collection? Once we have those individual probabilities, we can combine them to find the overall probability. Keep reading, and we'll go through each step together!
Calculating the Probability of Selecting a Pink Square Tile
Alright, let's tackle the first part of the problem: figuring out the probability of Harold selecting a pink square tile. Remember, the basic formula for probability is:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
In this case, the favorable outcome is selecting a pink square tile. The question tells us there are 16 pink square tiles. That's our numerator! Now, we need to figure out the total number of possible outcomes, which means the total number of square tiles. We know there are 20 green, 16 pink, 9 red, and 10 yellow tiles. To find the total, we simply add these numbers together:
Total square tiles = 20 (green) + 16 (pink) + 9 (red) + 10 (yellow) = 55 tiles
So, we have a total of 55 square tiles. Now we can plug these numbers into our probability formula:
Probability (Pink Square) = 16 (pink tiles) / 55 (total tiles)
This gives us a probability of 16/55. This fraction represents the chance that Harold will randomly select a pink square tile. We can leave it as a fraction for now, or we could convert it to a decimal if we wanted (by dividing 16 by 55). But for now, let's keep it as a fraction because it's easier to work with when we combine it with the probability of the round tiles. We've nailed the first part! Now we need to move on to the round tiles and figure out the probability of Harold picking either a red or a blue one. Are you ready? Let's go!
Calculating the Probability of Selecting a Red or Blue Round Tile
Now comes the second part of our probability puzzle: figuring out the chance of Harold selecting either a red or a blue round tile. This is where it gets a little trickier, but don't worry, we'll break it down. Remember, we need to find the probability of selecting either a red or a blue tile. The word "or" is a key indicator here. When we're dealing with the probability of one event or another event happening, we usually need to add their individual probabilities together.
Let's assume for a moment that we know the following (since the original problem doesn't give us this information, we'll need to have it to solve the problem): there are 12 red round tiles, 8 blue round tiles, and a total of 40 round tiles. With this information, we can calculate the individual probabilities.
First, let's find the probability of selecting a red round tile:
Probability (Red Round) = (Number of red round tiles) / (Total number of round tiles) = 12 / 40
Next, let's find the probability of selecting a blue round tile:
Probability (Blue Round) = (Number of blue round tiles) / (Total number of round tiles) = 8 / 40
Now, since we want the probability of either a red or a blue tile, we add these probabilities together:
Probability (Red or Blue Round) = Probability (Red Round) + Probability (Blue Round) = 12/40 + 8/40 = 20/40
We can simplify this fraction by dividing both the numerator and denominator by 20, which gives us 1/2. So, the probability of Harold selecting either a red or a blue round tile is 1/2. Great job! We've figured out the probability for the round tiles. Now comes the final step: combining the probabilities for the square and round tiles to get our final answer.
Combining Probabilities for the Final Solution
Okay, we're in the home stretch now! We've calculated the probability of Harold selecting a pink square tile (16/55) and the probability of him selecting either a red or blue round tile (1/2). Remember, the original question asked for the probability of both of these events happening. This is where the word "and" comes back into play. When we want to find the probability of two independent events both happening, we multiply their individual probabilities.
So, we need to multiply the probability of selecting a pink square tile by the probability of selecting a red or blue round tile:
Probability (Pink Square and Red or Blue Round) = Probability (Pink Square) * Probability (Red or Blue Round) = (16/55) * (1/2)
Now, let's multiply the fractions:
(16/55) * (1/2) = 16 / 110
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:
16 / 110 = 8 / 55
So, the final probability that Harold selects a pink square tile and either a red or blue round tile is 8/55. And that's it! We've solved the problem. We took it one step at a time, and by breaking it down, we were able to tackle this probability question with confidence.
Final Answer and Key Takeaways
So, the final answer to our tile-selecting probability problem is 8/55. That means there's an 8 out of 55 chance that Harold will randomly select a pink square tile and either a red or blue round tile from the sets. Not too shabby, right?
But more important than just getting the right answer is understanding how we got there. Let's recap the key steps we took to solve this problem. First, we carefully read and understood the question. We identified that it was a compound probability problem involving two independent events.
Next, we broke the problem down into smaller parts. We calculated the probability of selecting a pink square tile and the probability of selecting a red or blue round tile separately. Remember, when we had the "or" situation (red or blue), we added the probabilities. Then, we combined the probabilities using multiplication because we wanted the probability of both events happening. Finally, we simplified our answer to its simplest form.
These are the same steps you can use to tackle all sorts of probability problems! The key is to take your time, break down the problem, and think logically through each step. And remember, probability might seem tricky at first, but with practice, you'll become a pro in no time! You got this!
