Probability Of Three Knights Sitting Together

Let's dive into a probability problem that involves knights, a round table, and some strategic seating arrangements. Specifically, we're going to figure out the probability of selecting three knights, out of ten seated around a round table, such that all three are sitting next to each other. This is a classic problem that combines combinatorics and probability, requiring us to carefully consider the possible arrangements and selections.

Understanding the Problem

So, picture this: you've got ten knights chilling around a round table, probably after a long day of questing and dragon slaying. Now, we need to pick three of these valiant fellows at random. The big question is: what are the chances that the three knights we pick are sitting right next to each other? To crack this, we'll need to use a bit of combinatorics—counting the different ways this can happen—and then divide it by the total number of ways to pick any three knights.

Total Number of Ways to Choose Three Knights

First, let's figure out how many different ways we can pick any three knights out of the ten. This is a combination problem, since the order in which we pick the knights doesn't matter. We use the combination formula, which is:

C(n, k) = n! / (k!(n-k)!)

Where:

• n is the total number of items (in our case, knights).
• k is the number of items we're choosing (in our case, 3 knights).
• ! denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1).

So, for our problem, we have C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120. This means there are 120 different ways to pick three knights from the ten.

Number of Ways to Choose Three Adjacent Knights

Now, let's figure out how many ways we can pick three knights who are sitting next to each other. Since the table is round, we need to consider that the first and last knights are also adjacent. Think of it like this: we can start with any knight and then pick the two knights to their immediate right. Here’s how it breaks down:

1. Identify Possible Triplets: Because the knights are sitting in a circle, we can simply count the number of starting positions for a group of three consecutive knights. Since there are 10 knights, there are 10 possible starting positions for a group of three consecutive knights. For example, knights 1-2-3, 2-3-4, 3-4-5, and so on, up to 10-1-2.

2. Enumerate the Possibilities: Let's list them out to make it crystal clear:

   • Knights 1, 2, 3
   • Knights 2, 3, 4
   • Knights 3, 4, 5
   • Knights 4, 5, 6
   • Knights 5, 6, 7
   • Knights 6, 7, 8
   • Knights 7, 8, 9
   • Knights 8, 9, 10
   • Knights 9, 10, 1
   • Knights 10, 1, 2

So, there are 10 ways to choose three knights who are sitting next to each other.

Calculating the Probability

Alright, we're in the home stretch! Now that we know the total number of ways to pick three knights (120) and the number of ways to pick three adjacent knights (10), we can calculate the probability. The probability is the number of favorable outcomes (picking three adjacent knights) divided by the total number of possible outcomes (picking any three knights).

Probability = (Number of ways to choose three adjacent knights) / (Total number of ways to choose three knights)

Probability = 10 / 120 = 1 / 12

So, the probability that the three knights you pick are sitting next to each other is 1/12, or approximately 0.0833 (about 8.33%).

Conclusion

In summary, the probability of randomly selecting three knights sitting next to each other from a round table of ten is 1/12. This problem highlights how combinatorics and probability work together to solve real-world scenarios, even if that world involves knights and round tables. It's all about counting the possibilities and then figuring out the likelihood of the specific outcome you're interested in.

Key Concepts Revisited

Let's solidify our understanding by recapping the key concepts we used to solve this problem:

• Combinations: Understanding combinations (C(n, k)) is crucial. Combinations help us determine the number of ways to choose items from a larger set without regard to order. In our case, we used combinations to find out how many ways we could pick three knights out of ten.
• Probability: Probability is the measure of the likelihood that an event will occur. It’s calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
• Circular Arrangements: Recognizing that the knights were seated in a circle was key. This meant that the last knight was adjacent to the first, which affected how we counted the number of ways to select three adjacent knights.

Real-World Applications

While this problem might seem purely theoretical, the concepts we used are applicable in many real-world situations. Here are a few examples:

• Scheduling: Imagine you need to schedule three consecutive tasks out of ten available slots. The probability of those three tasks being scheduled consecutively can be calculated using similar methods.
• Quality Control: In a manufacturing process, you might want to test three consecutive items from a production line to check for defects. Calculating the probability of picking three defective items in a row can help you assess the quality of the process.
• Genetics: In genetics, you might be interested in the probability of finding three consecutive genes with a specific trait on a circular chromosome.

Additional Practice Problems

To further sharpen your skills, try solving these similar problems:

1. What is the probability of selecting four adjacent knights from a round table of twelve?
2. Suppose there are fifteen people sitting around a round table. What is the probability of selecting four people such that no two are sitting next to each other?
3. What if the knights are sitting in a row instead of around a table? How does this change the probability calculation?

By working through these problems, you’ll gain a deeper understanding of combinatorics and probability, and you’ll be better equipped to tackle similar challenges in the future.

So, there you have it! The next time you're planning a medieval feast, you'll know exactly how to calculate the odds of your chosen knights sitting together. Happy problem-solving, guys!