Recycling Probabilities: A Step-by-Step Guide

Hey guys! Ever wondered how math concepts can be applied to everyday situations like school recycling drives? Well, let's dive into an interesting scenario where we'll explore probabilities related to a recycling drive. We'll break down how to calculate the chances of picking a plastic bottle from the bin and the likelihood that it was recycled by a tenth-grader. Buckle up, because we're about to make math fun and relevant!

Understanding the Basics of Probability

Before we jump into the specifics of our recycling problem, let's quickly refresh our understanding of probability. Probability, at its core, is simply the measure of how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Think of it as a way to quantify uncertainty.

• Key Concepts:

  • Event: An event is a specific outcome or set of outcomes that we're interested in. For example, in our recycling drive, the event could be picking a plastic bottle.

  • Sample Space: The sample space is the set of all possible outcomes. In the recycling context, the sample space would include all the items in the recycling bin – plastic bottles, aluminum cans, paper, etc.

  • Probability Formula: The probability of an event (let's call it A) is calculated as:

    P(A) = (Number of favorable outcomes for A) / (Total number of possible outcomes in the sample space)

To really nail this down, imagine flipping a fair coin. There are two possible outcomes: heads or tails. The probability of getting heads is 1 (favorable outcome) divided by 2 (total possible outcomes), which equals 0.5 or 50%. See? Probability isn't so scary after all! Now, let's apply these concepts to our recycling drive.

Setting Up the Recycling Scenario

Okay, let’s paint a picture. Imagine your school has organized a fantastic recycling drive! Students from different grades have been diligently tossing in their recyclable items. Now, to analyze the data, we've got a table showing the breakdown of items collected. This table is our key to unlocking the probabilities we're interested in. Let's assume our table looks something like this:

This table gives us a clear view of how many of each item were recycled by students from each grade. The “Total” row and column are crucial because they provide the overall numbers we'll need for our calculations. Now, let's define the events we're interested in:

• Event A: The item pulled out of the recycling bin is a plastic bottle.
• Event B: A tenth-grader recycled the item.

Our goal is to figure out the probability of each of these events, both individually and potentially in combination. So, let’s get started with calculating P(A), the probability of picking a plastic bottle.

Calculating P(A): Probability of Picking a Plastic Bottle

Alright, let's tackle the first part: finding the probability of picking a plastic bottle (Event A). Remember the probability formula we discussed earlier? P(A) = (Number of favorable outcomes for A) / (Total number of possible outcomes in the sample space). In our case:

• Favorable outcomes for A: This is the number of plastic bottles in the recycling bin. Looking at our table, we can see there are a total of 120 plastic bottles.
• Total number of possible outcomes in the sample space: This is the total number of all items in the recycling bin. From the table, we see a grand total of 370 items.

Now, we just plug these numbers into our formula:

P(A) = 120 / 370

We can simplify this fraction by dividing both the numerator and denominator by 10:

P(A) = 12 / 37

To get a better sense of this probability, we can convert it to a decimal by dividing 12 by 37, which gives us approximately 0.324. If we want to express this as a percentage, we multiply by 100, resulting in about 32.4%.

So, the probability of randomly picking a plastic bottle from the recycling bin is approximately 32.4%. That means if you were to reach into the bin without looking, you'd have roughly a one-in-three chance of grabbing a plastic bottle. Pretty cool, huh? Now, let’s move on to calculating the probability related to tenth-graders.

Calculating P(B): Probability a Tenth-Grader Recycled the Item

Next up, we're figuring out the probability that a tenth-grader recycled the item (Event B). We'll use the same probability formula, but this time, our focus shifts to the tenth-graders' contributions:

P(B) = (Number of items recycled by tenth-graders) / (Total number of all items)

Let's break down the numbers:

• Number of items recycled by tenth-graders: To find this, we need to look at the