Anna And Bastien's Rendezvous: A Probability Puzzle

Hey guys! Let's dive into a fun probability problem involving Anna and Bastien's rendezvous. This is Exercise 4, where we'll explore the likelihood of Anna and Bastien meeting up at Charlotte's place. Understanding this problem is a great way to grasp the concepts of uniform distribution and independent random variables. So, grab a coffee (or whatever you like) and let's get started! We will explore the timing of their arrivals and use mathematical tools to figure out the chances of them being at Charlotte's at the same time. This is a classic example of probability in action, and it’s a good one for sharpening your skills in this area. We’ll break down the scenario step by step, making it easy to understand even if you're not a math whiz. In this context, the entire 12 PM to 2 PM window is considered. The variables X and Y represent Anna and Bastien's arrival times, respectively, with '0' corresponding to 12 PM and '2' corresponding to 2 PM. Each of their arrival times is assumed to follow a uniform distribution within this two-hour window. This means that any arrival time within the interval [0, 2] is equally likely. The question we're tackling is centered around the concept of probability. Specifically, we want to figure out the chance that Anna and Bastien's arrival times overlap during their two-hour window at Charlotte's place. This kind of problem is very common in probability theory and can be applied to a variety of real-world situations, from scheduling meetings to predicting when two events might coincide. We'll be using the properties of uniform distributions and independence to get to the answer, so let's get into the specifics of the scenario and how we approach it mathematically.

Understanding the Setup: Uniform Distribution and Independence

Alright, let's break down the key concepts here. First off, what does a uniform distribution mean in this context? Imagine a perfectly fair dice where each number has an equal chance of being rolled. In our scenario, the arrival times of Anna and Bastien are like that dice roll. Any time between 12 PM (0) and 2 PM (2) is equally probable for both of them. This is what we mean by a uniform distribution on the interval [0, 2]. Picture a straight line graph: the probability is the same across the entire range. Then, we have independent random variables. What does it mean? Anna's arrival time doesn't affect Bastien's, and vice versa. It’s like flipping two coins; the result of one doesn't influence the other. This independence is a crucial aspect of the problem, and simplifies things a lot. So, Anna and Bastien's arrivals are independent and uniformly distributed over the interval [0, 2]. The interval [0, 2] is our timeframe (12 PM to 2 PM), with each moment being equally likely for their arrival. The fact that the variables are independent means there's no correlation; Anna's arrival doesn’t give us any clue about Bastien's. This simplifies the math and allows us to use specific formulas and methods to calculate the probability they meet. This setup is a classic example, helping us to apply probability principles to a practical problem, in this case, a real-life situation of someone arriving at the same location.

Calculating the Probability of a Meeting

So, how do we calculate the probability that Anna and Bastien meet? The key is to consider the conditions under which they will meet. They will meet if their arrival times overlap within a certain time frame. For example, if Anna arrives at 12:30 PM (0.5 on our scale) and Bastien arrives at 1:00 PM (1.0), they've met. But if Anna arrives at 12:05 PM (0.08) and Bastien at 1:55 PM (1.92), they haven't. The question translates into figuring out the area of the region where |X - Y| ≤ some time constraint. In this case, we have a uniform distribution, and the key is to visualize the problem on a 2D graph with X and Y axes. The region of interest is where the difference between X and Y is within a certain time, which can be expressed in terms of the problem requirements. This area then represents the probability of the event occurring. We need to find the area of this region within the square [0, 2] x [0, 2]. Let’s get into the details: The area of the entire square representing all possible arrival times is 2 * 2 = 4 (since the range of each variable is from 0 to 2). We then need to figure out the area where the absolute difference between X and Y is less than or equal to a certain value. Once we have this area, we divide it by the total area (4) to get the probability. This is where the visualization on the graph comes in handy. You can graph this on a piece of paper or with a software program. Drawing the lines on your graph helps visualize what is happening. Each axis is an arrival time and the area where the difference between them is less than or equal to a value, this is the area where they meet. Calculating this probability involves a little bit of geometry, and understanding how independent variables work. It's a fun example of how abstract mathematical ideas can model real-world scenarios.

Visualizing the Solution with a Graph

To make this clearer, let's visualize this with a graph. Imagine a square with sides of length 2 (representing the time frame from 0 to 2 for both Anna and Bastien). The x-axis represents Anna's arrival time (X), and the y-axis represents Bastien's arrival time (Y). Each point within this square represents a possible combination of arrival times for Anna and Bastien. The total area of the square is 4. Now, if they have to meet, the question becomes: how do we define the region where they meet? If we want them to meet within a specific time, it means the absolute difference between their arrival times (|X - Y|) must be less than or equal to a time threshold, let's say 't'. This condition is crucial. On the graph, this condition defines a band. If we assume they must meet within 30 minutes (0.5 on our scale), we're interested in the area of the region where |X - Y| ≤ 0.5. This area is the region within the square that lies between the lines Y = X + 0.5 and Y = X - 0.5. These lines are parallel to the diagonal of the square (where X = Y). The area where they don’t meet will be the two triangles that are cut off from the main square by these lines. The two triangles will be at each corner of the square. The area of each triangle can be calculated as 0.5 * base * height. The base and height of each triangle will be equal to each other in this setup. To calculate the probability, subtract the area of these two triangles from the total area of the square and divide by the total area. This provides a clear, visual representation of the problem, turning an abstract math problem into something more concrete. This visual approach is a great tool for understanding probability questions.

Step-by-Step Calculation and Probability

Alright, let’s do the step-by-step calculation. We have the total area of the square is 4. Now, let’s calculate the area where they do not meet. Let's assume the time window they have to meet within is 30 minutes, or 0.5 hours. The two triangles that represent the