Model Answer
0 min readIntroduction
Probability, at its core, is the measure of the likelihood of an event occurring. In real-world scenarios, probabilistic reasoning is crucial for decision-making under uncertainty. This problem presents a classic example of geometric probability, where the probability of an event is determined by the ratio of areas. The scenario involves two individuals, A and B, attempting to meet within a specified time frame, each willing to wait for a limited duration. Understanding the constraints and visualizing the possible arrival times is essential to determine the probability of a successful meeting.
Understanding the Problem
Let's represent the time interval between 3 p.m. and 4 p.m. as 60 minutes. We can consider 'x' as the arrival time of A (in minutes after 3 p.m.) and 'y' as the arrival time of B (also in minutes after 3 p.m.). Both x and y lie between 0 and 60. The condition for them to meet is that |x - y| ≤ 10. This means A arrives within 10 minutes of B, or B arrives within 10 minutes of A.
Geometric Representation
We can visualize this problem graphically. Consider a square in the x-y plane where 0 ≤ x ≤ 60 and 0 ≤ y ≤ 60. The total area of this square represents all possible arrival time combinations for A and B, which is 60 * 60 = 3600 square minutes.
Defining the 'Success' Region
The condition |x - y| ≤ 10 can be rewritten as -10 ≤ x - y ≤ 10, which gives us two inequalities:
- y ≥ x - 10
- y ≤ x + 10
Calculating the Area of the 'Success' Region
The area of the region where they *don't* meet consists of two right-angled triangles.
- Triangle 1: y < x - 10. This triangle has vertices (10, 0), (60, 0), and (60, 50). Its area is (1/2) * 50 * 50 = 1250.
- Triangle 2: y > x + 10. This triangle has vertices (0, 10), (0, 60), and (50, 60). Its area is (1/2) * 50 * 50 = 1250.
Calculating the Probability
The probability of A and B meeting is the ratio of the area of the 'success' region to the total area: Probability = (Area of 'success' region) / (Total area) = 1100 / 3600 = 11/36.
Alternative Approach - Considering Waiting Time
Another way to think about this is to consider the effective time window for each person. If A arrives at time 'x', B must arrive between x-10 and x+10. Similarly, if B arrives at time 'y', A must arrive between y-10 and y+10. The probability calculation remains the same, but this approach helps visualize the constraints.
Conclusion
Therefore, the probability that A and B meet, given their agreed-upon time frame and waiting limit, is 11/36. This problem demonstrates a practical application of geometric probability, where visualizing the problem space and calculating areas provides a clear solution. The key takeaway is understanding how constraints affect the possible outcomes and how to represent them mathematically.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.