Probability Of Drawing $1 And $5 Bills: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever find yourself wondering about the probability of pulling specific bills out of your wallet? Let's break down a fun mathematical problem that tackles just that. We'll walk through it step-by-step so you can confidently calculate these scenarios yourself. Imagine you've got a wallet stuffed with cash – four $20 bills, ten $5 bills, and twelve $1 bills. Now, what's the chance you'll grab a $1 bill on your first try and then a $5 bill on your second try? Sounds like a fun puzzle, right? Let's dive in and figure it out together!

Understanding the Problem: Bills and Probabilities

So, let's clearly define the problem we're tackling. We need to determine the probability of two events happening in sequence: first, drawing a $1 bill, and second, drawing a $5 bill. Remember, probability is essentially the chance of a specific event occurring, expressed as a fraction. The numerator is the number of favorable outcomes (the outcomes we're interested in), and the denominator is the total number of possible outcomes.

In our case, the favorable outcomes are drawing a $1 bill first and then a $5 bill. The total possible outcomes are all the different combinations of bills we could draw in two tries. Before we jump into calculations, let's take a good look at what's in our wallet. We've got a mix of bills, and that mix is going to affect the probabilities. We have four $20 bills, which are great for larger purchases. We also have ten $5 bills, perfect for grabbing a quick bite or a coffee. And finally, we have twelve $1 bills, useful for smaller transactions or making exact change. This combination of bills gives us a good foundation for calculating our probabilities. The trick is to consider each step separately and then combine the probabilities to get our final answer.

Step 1: Probability of Drawing a $1 Bill First

Okay, guys, let's start with the first part of our problem: the probability of drawing a $1 bill on the first try. To figure this out, we need to know two things: how many $1 bills we have and the total number of bills in the wallet. We already know we have twelve $1 bills. Now, let's calculate the total number of bills. We have four $20 bills, ten $5 bills, and twelve $1 bills. Adding those up (4 + 10 + 12), we get a total of 26 bills. So, we have 12 favorable outcomes (drawing a $1 bill) out of 26 total possible outcomes (drawing any bill). Therefore, the probability of drawing a $1 bill first is 12/26. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us a simplified probability of 6/13. Remember this fraction – it's the first piece of our puzzle! Understanding this initial probability is crucial because it sets the stage for the second part of our calculation. We've established the likelihood of the first event, and now we need to consider how that affects the probability of the second event. It's like setting up the first domino in a chain reaction – it influences what happens next.

Step 2: Probability of Drawing a $5 Bill Second (After Drawing a $1 Bill)

Now, this is where it gets a little trickier, but don't worry, we've got this! We need to figure out the probability of drawing a $5 bill second, but here's the catch: we've already taken a $1 bill out of the wallet. This means the total number of bills in the wallet has decreased, and the number of $1 bills has also decreased. So, how does this change things?

Well, since we removed one bill, the total number of bills in the wallet is now 25 (26 - 1). We still have ten $5 bills in the wallet because we didn't take one of those out. So, the number of favorable outcomes for drawing a $5 bill is still 10. Therefore, the probability of drawing a $5 bill second, after taking out a $1 bill, is 10/25. We can simplify this fraction as well by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us a simplified probability of 2/5. This step highlights the importance of considering the impact of previous events on subsequent probabilities. The act of drawing the first bill changes the composition of the wallet and, consequently, the chances of drawing specific bills in the future. This is a key concept in probability, especially when dealing with dependent events.

Step 3: Combining the Probabilities

Alright, folks, we've got the probability of each event happening separately. Now, we need to combine them to find the probability of both events happening in sequence. This is where the multiplication rule of probability comes into play. The multiplication rule states that the probability of two independent events both occurring is the product of their individual probabilities. In our case, the events are not entirely independent because the first draw affects the second draw (as we saw in step 2). However, the principle of multiplying probabilities still applies, taking into account the changed conditions after the first draw. We calculated the probability of drawing a $1 bill first as 6/13. We then calculated the probability of drawing a $5 bill second (after taking out a $1 bill) as 2/5. To find the probability of both these events happening, we multiply these two fractions together: (6/13) * (2/5). Multiplying the numerators (6 * 2) gives us 12. Multiplying the denominators (13 * 5) gives us 65. So, the combined probability is 12/65. This final calculation ties together all the previous steps, demonstrating how individual probabilities can be combined to determine the likelihood of a series of events. It's a powerful tool for analyzing various scenarios, from drawing bills from a wallet to predicting outcomes in games of chance.

Final Answer: The Probability of Drawing a $1 and then a $5

So, after all that calculation, we've arrived at our final answer! The probability of drawing a $1 bill first and then a $5 bill is 12/65. This means that if you were to reach into your wallet twice, a considerable number of times, you would expect to draw a $1 bill followed by a $5 bill approximately 12 out of every 65 times. Isn't that neat? This problem demonstrates the power of probability in predicting outcomes in everyday situations. While the result might not be immediately intuitive, breaking it down into smaller steps makes the logic clear and the solution achievable. Plus, understanding these principles can help you make more informed decisions in various aspects of life, from financial investments to even simple games. Now you can impress your friends with your probability prowess! And remember, math can be fun and practical – it's all about understanding the underlying concepts and applying them creatively.