Calculating Probabilities: Mangoes, Apples & Spoiled Fruits

Hey guys! Let's dive into a fun probability problem! This one involves mangoes, apples, and a little bit of spoilage. We'll break down the steps to figure out the chances of picking specific fruits from two different bags. Get ready to flex those math muscles! The problem involves calculating the probability of selecting certain fruits from two different bags, one containing mangoes and the other containing apples. It's a classic example of how probability and combinations work together. Let's go through this step by step and make sure we understand every single detail. This will give you a solid understanding of how to tackle similar problems in the future. Get your thinking caps on, because we're about to make sense of this probability puzzle together!

Understanding the Problem: The Setup

Okay, so here’s the scenario: We've got two bags. Bag A has 8 mangoes, and, unfortunately, 2 of them are rotten. Bag B has 10 apples, with 5 of them being less than perfect. The goal? To figure out the probability of grabbing 4 mangoes and 3 apples, given the conditions. The key here is recognizing that we’re dealing with two separate events: picking mangoes and picking apples. And for each, we need to calculate the odds of getting exactly the right mix of good and bad fruit. Think of it as two mini-problems rolled into one. You’ll need to understand how to calculate combinations to solve this properly. Let's first break down each bag and figure out the number of combinations of mangoes and apples that meet the criteria. It is important to ensure we are on the right track. So, let's get into the details.

First things first, let's look at the mangoes. Bag A has 8 mangoes total, with 2 spoiled ones. That means 6 are good. We want to pick 4 mangoes total. The good news is that we don't care if they are good or bad, we just need 4 of them. The same goes for the apples. We need to consider the combinations of both the good and bad mangoes to calculate the probability of selecting 4 mangoes.

Moving on to the apples. Bag B has 10 apples, 5 of which are spoiled. We need to pick 3 apples in total. And again, we don't mind if the apples are good or bad, but calculating the combinations of both good and bad apples will help calculate the probability of selecting 3 apples. Each step, from the mangoes in Bag A to the apples in Bag B, contributes to the overall probability, which requires using combination formulas to calculate the exact number of ways to select the fruit according to the conditions. The next step is putting all of it together and crunching some numbers. Sounds interesting right? Let's keep going!

Calculating Mango Probabilities: Bag A

Alright, let's focus on Bag A, the mango bag. We need to figure out the number of ways we can select 4 mangoes from the total 8. We'll use combinations for this since the order we pick the mangoes in doesn’t matter. Remember that a combination is about selecting items from a set without regard to the order. The formula for combinations is: C(n, k) = n! / (k! * (n-k)!) where n is the total number of items, k is the number of items we want to choose, and '!' means factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1). For the mangoes, n = 8 (total mangoes) and k = 4 (mangoes we want to pick). So, we have C(8, 4) = 8! / (4! * 4!) = (8765) / (4321) = 70. This means there are 70 different ways to choose 4 mangoes from the 8 in Bag A.

Now, let's think about this from another angle. Let's consider the specifics of the problem: since we are not concerned about the quality of the mangoes, it's as simple as using the formula to calculate the possible combinations of selecting 4 mangoes out of 8. But, if the problem specified that we had to select a certain number of spoiled and good mangoes, we'd have to break down the calculation further. For example, if the question was what is the probability of picking 2 good mangoes and 2 spoiled mangoes from Bag A? Then, we would calculate the combinations separately for both good and bad mangoes, and then multiply the combinations together. But, in this scenario, we simply need to calculate the combinations of picking 4 mangoes out of 8. So, the total possible outcomes when picking 4 mangoes from Bag A is 70.

Keep in mind, the number 70 gives us the total possible combinations when selecting 4 mangoes. This total needs to be further adjusted by calculating the combinations for the apples in bag B. The formula can be a little tricky at first, but as we keep solving these problems, you will have no problem in understanding the concepts.

Calculating Apple Probabilities: Bag B

Time to switch gears and head over to Bag B, the apple haven! We’ve got 10 apples here, and we want to pick 3. Again, we're using combinations because the order of selection doesn't matter. Using the combinations formula: C(n, k) = n! / (k! * (n-k)!), where n = 10 (total apples) and k = 3 (apples we want to pick). So, C(10, 3) = 10! / (3! * 7!) = (1098) / (321) = 120. This tells us there are 120 different ways to choose 3 apples from the 10 in Bag B.

Now, let's reflect on this. Like with the mangoes, the 120 represents the total number of combinations when picking 3 apples from the 10. If the questions had specified that we needed to select only good apples, or specific combinations of good and bad apples, we would calculate these separately. It is very important to fully understand what the questions are asking to give the right answer. In this case, however, the question only asks us to calculate the combinations of picking 3 apples in total. So, the total outcomes for picking 3 apples from Bag B is 120.

Now we’ve calculated the combinations for both the mangoes and the apples individually. We've got the groundwork laid out, and we're ready to get to the fun part. We have all the numbers we need to determine the probability. But before we combine them, it's worth noting that these calculations highlight the power of combinations in probability. They help us account for every possible selection, which ensures an accurate result. Let's combine all of this to get the final probability of picking mangoes and apples from both bags.

Combining Probabilities: The Final Calculation

Alright, we're in the home stretch! We've figured out the possible outcomes for the mangoes and the apples separately. Now, we need to put it all together to find the probability of picking 4 mangoes AND 3 apples. Since the selection from Bag A is independent of the selection from Bag B, we multiply the number of combinations from each bag together. Think of it like this: for every way you can pick mangoes, there are several ways to pick apples, and we have to account for all of them.

For the mangoes, we found 70 possible combinations. For the apples, we found 120 possible combinations. So, the total number of combinations for picking both mangoes and apples is 70 * 120 = 8400. This means there are 8400 ways to pick the fruits according to our criteria, that is 4 mangoes and 3 apples.

To find the total number of possible outcomes, we must account for the number of combinations to select 7 items (4 mangoes and 3 apples) from both bags. Here, we add the total number of mangoes and apples from both bags and calculate the combination from the totals. We have 8 mangoes and 10 apples, adding up to 18 total fruits. The number of possible combinations to select 7 fruits from 18 is: C(18, 7) = 18! / (7! * 11!) = (18171615141312) / (7654321) = 31,824. We have determined the possible combinations for picking both mangoes and apples (8400), and we have also determined the total possible outcomes (31,824).

Finally, to find the probability, we divide the favorable outcomes (picking 4 mangoes and 3 apples) by the total possible outcomes (picking any 7 fruits). Therefore, the probability is: 8400 / 31,824 = 0.264. Which rounds off to 0.26.

So, the probability of picking 4 mangoes and 3 apples is approximately 0.26.

Final Thoughts: Putting it All Together

And there you have it, guys! We have successfully navigated a probability problem involving mangoes, apples, and spoilage. We broke down the problem into manageable steps, applied the combinations formula, and calculated the final probability. Remember, the key to solving these problems is understanding the concepts of combinations and how to apply them. Breaking down complex problems into smaller parts makes them more manageable, and with practice, you'll become a pro at calculating probabilities. Always remember to read the question carefully, consider the conditions, and plan your steps. Keep practicing, and you'll master these kinds of problems. Congrats on making it through! Keep up the great work, and keep exploring the exciting world of probability and mathematics!