Over a year ago I heard statistics professor Allan Rossman speak during a virtual session. The problem he shared was easy to understand and potentially challenging to solve.
Here's the scenario:
Two friends agree to meet for lunch. They agree to wait for 15 minutes for the other person to arrive. If they don't arrive in that 15 minute window, then they leave. The friends arrive at a random time between noon and 1 PM. What is the probability they will see each other?
Allan demonstrated how to solve this problem using a simulation in R and I decided to try creating a simulation for the problem using Desmos. I presented my Desmos version of this solution during a 15 minute Epsilon Talk at Moravian University during the spring semester. This blog will take you through that talk.
Let's consider the following times. Do they work or not?
To help us think about all the different random rendevous pairings, we will use the following definition.
Let (A, B) represent the minutes after noon that Person A arrives and the minutes after noon Person B arrives.
Example: Person A arrives at 12:15 PM and Person B arrives at 12:56 PM. This would be represented as (15, 56).
Here's an image that shows many ordered pairs like (15, 56). Which pairs correspond to meeting for lunch?
Remember that there is a 15 minute window were each person will wait for the other person. This means
and we can color code our Desmos graph, where red dots mean that the two friends met and blue dots means the friends did not meet. Is there a pattern that emerges?
I invite you to go to this
Desmos calculator page and use the slider to investigate adding more random ordered pairs. What shape is formed by the red dots?
Other Desmos simulations can be seen
here and
here. The first simulation creates a histogram with the width of each bar being 15 minutes. The second simulation uses a uniform distribution and a ticker and was created by Andrew Knauft.
Of couse any good question leads to more questions...here are some others that I invite you to explore:
How long should the friends wait if they want the probability of meeting for lunch to be 50%?
What if they wait different amounts of time?
What if they arrive randomly in a 30 minute window of time?
What if more than 2 people want to meet for lunch?
Please share your answers to these questions or your own random rendevous questions.
