Two friends arrange for a lunch date between 12:00 and 1:00. A week later, however, neither of them remembers the exact meeting time. As a result, each person arrives at a random time between 12:00 and 1:00 and waits exactly 10 minutes for the other person. When the 10 minutes have passed, each person leaves if the other person has not come. What is the probability the friends will meet?

This isn’t an easy question, but it is open to a beautiful geometric interpretation that makes it much easier to understand and solve. Using Web Sketchpad, you can do the exploration directly in your browser. Check out the interactive model below (or here).

Here is some information about using the model that you’ll find helpful:

Point A represents one friend; point B represents the other friend. The locations of the points along the two axes represent the times when the friends arrive.

Point P represents the two arrival times as a single point. Notice that in its original location, point P is green. This means that A and B arrive within 10 minutes of each other and successfully meet up.

Press the button Run the Simulation Once. Points A and B will move to new, random locations. Point P will remain green if the friends’ arrival times are within 10 minutes of each other, but point P will turn red if the friends do not arrive in time to meet. Press the button several timesto make sure you understand when point P is green and when it is red.

Notice that point P leaves a trace behind of all its prior locations. By running the probability simulation multiple times, you can see whether there’s a pattern to be discerned in the placement of the green and red points. Press Run the Simulation Repeatedly, grab a coffee, sit back, and watch the screen fill with color.

How can you use the pattern of green and red points to determine the likelihood that the two friends will meet?

To clear the traces of the points, press Start Again. To explore the question for different waiting times, drag point T around the circle. For example, when T is at 30, you can examine the likelihood that the two friends will meet if each is willing to wait 30 minutes for the other to arrive.