Probability: Dimes & Quarters Without Replacement

Hey guys! Let's dive into a probability problem that might seem a bit tricky at first, but I promise we'll break it down step by step so it becomes super clear. We're going to tackle a classic scenario involving drawing coins from a bag without replacing them. This "without replacement" part is crucial because it affects the probabilities as we go. So, grab your thinking caps, and let's get started!

Understanding the Problem

Let's get crystal clear on what we're trying to figure out. Our main goal is to calculate the probability of a specific sequence of events: first, drawing a dime, and then, without putting that dime back, drawing a quarter. We've got a bag filled with a mix of coins – 2 quarters, 3 dimes, and 4 pennies. The key here is that once we take a coin out, it's gone, and that changes the total number of coins and the number of each type of coin left in the bag. This is what makes it a without replacement problem, and it's a bit different from situations where you put the coin back in before the next draw.

To really nail this, we need to think about each draw as a separate event and then combine their probabilities. We'll use some fundamental probability concepts, like the probability of an event being the number of favorable outcomes divided by the total number of possible outcomes. Sounds like a mouthful, but it's pretty straightforward once we apply it. We'll also need to remember that when we want to find the probability of two events happening in sequence (like drawing a dime and then a quarter), we'll often multiply their individual probabilities. So, with these ideas in mind, let's roll up our sleeves and get into the nitty-gritty of the calculations!

Breaking Down the First Draw: Probability of Drawing a Dime

Alright, let's zoom in on the very first action: reaching into the bag and pulling out a coin. Specifically, we're interested in the probability of grabbing a dime. To figure this out, we need to know two things: how many dimes are there, and how many total coins are in the bag at the start. According to the problem, we have a bag that contains 3 dimes. These are our favorable outcomes – the coins that, if drawn, would make our first event a success. Now, let's count up the total number of coins. We have 2 quarters, 3 dimes, and 4 pennies. Add those up, and we've got a grand total of 9 coins in the bag. This is the total number of possible outcomes when we make our first draw.

So, how do we translate this into a probability? Remember, probability is just the ratio of favorable outcomes to total possible outcomes. In this case, that's 3 dimes (favorable outcomes) divided by 9 total coins (possible outcomes). This gives us a probability of 3/9. But, we can simplify this fraction! Both 3 and 9 are divisible by 3, so we can reduce 3/9 to 1/3. This means there's a 1 in 3 chance of drawing a dime on our first try. We've nailed the first part of the problem! But don't get too comfy yet, because the next draw is where things get a little more interesting since we're not putting the dime back in the bag.

The Second Draw: Probability of Drawing a Quarter After Removing a Dime

Okay, so we've successfully snagged a dime from the bag and now we're setting it aside. This is where the "without replacement" part really kicks in. Because we didn't put the dime back, the total number of coins in the bag has changed. It's gone down by one! We started with 9 coins, and now we only have 8 left. This is a crucial detail because it affects the probability of our next event: drawing a quarter.

Now, let's think about the quarters. At the beginning, we had 2 quarters in the bag. Since we haven't drawn a quarter yet, that number hasn't changed. We still have 2 quarters sitting in the bag, waiting to be drawn. So, when we reach in for our second draw, we have 2 favorable outcomes (the 2 quarters) and 8 total possible outcomes (the remaining coins). This means the probability of drawing a quarter on the second draw is 2/8. Just like before, we can simplify this fraction. Both 2 and 8 are divisible by 2, so we can reduce 2/8 to 1/4. So, the probability of drawing a quarter after we've already taken out a dime is 1/4. We're getting closer to the final answer, but we're not quite there yet. We've got the probability of each individual event, but now we need to combine them to find the probability of both events happening in sequence.

Combining Probabilities: Finding the Probability of Both Events

We've figured out the probability of drawing a dime first (1/3) and the probability of drawing a quarter second (1/4), considering we didn't replace the dime. Now comes the crucial step: how do we combine these individual probabilities to find the overall probability of both events happening in the specific order we want – dime first, then quarter? This is where a fundamental rule of probability comes into play. When we want to find the probability of two independent events happening one after the other, we multiply their individual probabilities.

Think of it like this: each event is like a hurdle we need to clear. The probability of clearing both hurdles is a product of how likely we are to clear each one individually. So, in our coin-drawing scenario, we're going to multiply the probability of drawing a dime (1/3) by the probability of drawing a quarter after the dime has been removed (1/4). Let's do the math: (1/3) * (1/4) = 1/12. And there you have it! The probability of drawing a dime and then a quarter, without replacement, is 1/12. It's like we've solved a little puzzle, piece by piece, and now we have the satisfying final answer.

Checking Our Answer Against the Given Options

Alright, we've calculated the probability of drawing a dime and then a quarter, without replacement, and we've arrived at the answer 1/12. Now, let's make sure our hard work pays off and see if this answer matches any of the options provided in the original problem. Looking back at the options, we see: A. 2/27, B. 1/3, C. 5/17, D. 1/12. And there it is! Option D, 1/12, perfectly matches our calculated probability. This is super satisfying because it confirms that we've gone through the problem correctly, step by step, and arrived at the right solution. It's always a good idea to double-check your answer against the given options, especially in a multiple-choice scenario, just to make sure you haven't made any sneaky errors along the way. So, we can confidently say that the correct answer to this probability problem is indeed D. 1/12.

Key Takeaways

So, what have we learned from this coin-filled adventure? Let's recap the main takeaways so we can add them to our problem-solving toolkit. First and foremost, we tackled a "without replacement" probability problem. This means that the outcome of the first event (drawing a dime) directly impacted the probabilities of the subsequent event (drawing a quarter). Remember, when you're dealing with "without replacement" scenarios, the total number of outcomes and the number of favorable outcomes change after each event. This is a crucial detail to keep in mind!

We also reinforced the fundamental principle of calculating probability: it's the number of favorable outcomes divided by the total number of possible outcomes. This simple fraction is the foundation of probability calculations, and it's essential to understand it inside and out. Furthermore, we practiced combining the probabilities of sequential events. When we wanted to find the probability of drawing a dime and then a quarter, we multiplied their individual probabilities. This "multiplication rule" is a powerful tool for calculating the probability of multiple events happening in a specific order.

Finally, we emphasized the importance of breaking down complex problems into smaller, manageable steps. By focusing on one draw at a time and then combining the results, we transformed a seemingly tricky problem into a series of straightforward calculations. So, next time you encounter a probability problem, remember these takeaways: pay attention to "without replacement" scenarios, use the basic probability formula, apply the multiplication rule for sequential events, and break down the problem into smaller chunks. With these strategies in your arsenal, you'll be able to tackle probability challenges with confidence!