Data Science Interview Prep: Q65

The Waiting Game: A Visual Puzzle in Probability. (Category: Probability)

Romeo and Juliet agree to meet at a specific time. Each arrives at a random time uniformly between 0 and 1 hour after the agreed time. The first to arrive will wait for 15 minutes before leaving if the other hasn’t arrived. What is the probability that they meet?

Solution:

Problem Overview:

Romeo and Juliet agree to meet at a specific time. However, each may arrive at any random time within the hour after the agreed time, with all arrival times equally likely. Whoever arrives first will wait no more than 15 minutes before leaving if the other hasn’t yet arrived.

So, what’s the probability that they meet?

Let’s break it down visually.

We can represent Romeo’s and Juliet’s arrival times as coordinates on a square. In Figure 1 the horizontal axis shows Romeo’s arrival time (in hours), and the vertical axis shows Juliet’s arrival time (in hours). Every point inside the square represents a possible combination of arrival times.

The upper blue line in Figure 1 represents the combinations where Juliet arrives exactly 15 minutes after Romeo—beyond this line, she arrives too late and Romeo will have already left.

The lower blue line in Figure 1 represents the combinations where Romeo arrives exactly 15 minutes after Juliet—beyond this, he arrives too late and Juliet will no longer be waiting.

The shaded region between these two lines includes all the points where the difference in arrival times is 15 minutes or less—meaning they meet.

The total area of the entire square is 1 (since both arrival times range from 0 to 1 hour). The region where they will meet is the band between the two blue lines.

This meeting region is the area of the entire square minus the areas of the two unshaded right triangles—each triangle has an area of 1/2 × base × height, i.e.,

Hence, the area of the meeting region, i.e., the shaded region, is:

Therefore, the probability that Romeo and Juliet will meet is 7/16.

This problem illustrates the connection between geometry and probability, showing how examining the arrangement of possible outcomes helps calculate the chance of an event precisely.

We hope you found the article both enlightening and valuable! For more insightful content delivered straight to your inbox, simply enter your email address below and hit the subscribe button. Stay tuned for future updates!

Reference:

• Romeo Juliet Date Meetup: https://math.stackexchange.com/questions/1279873/basic-probability-romeo-and-juliette-meet-for-a-date