I met Ms. Samantha Griffith through the Kentucky Academy of Science conference early this November. The Computer Science room, where I had presented a research topic of my own, had left for lunch. Curious, I wandered into the Mathematics room and caught her presentation.
She begins with a problem brought to her by a local student organization—they’re hanging posters to advertise the club and want to know the locations that will maximize the probability of students joining and becoming active members.
She sets out to solve this problem as part of a research requirement. She walks us through the models she and her team developed throughout the semester, explaining the motivations and difficulties of each. Finally, she arrives at a brilliantly simple solution; unable to determine the numbers, however, that fit into this model, she concludes her presentation.
She notes that with the final steps still unsolved, her professor plans to lock away the unfinished research until another student comes along to work out Griffith’s numbers.
Joining and Becoming. Her first approach to tackling this club’s problem involves a complicated graph. Students interact in various ways with both one another and poster locations. Students are split into three groups: those in the club, those contacted about the club, and those the club still needs to target. Poster locations are similarly split between buildings.
Each interaction would have a certain probability of moving a “target student” one step closer to joining the club. Different poster locations in different buildings would have different probabilities, making the problem a simple “guess and check.” However, this model relies too heavily on which buildings students visit and who they talk to.
Brilliantly Simple. This complication prompts Griffith and her team to seek simplification. They remove buildings and student interaction from their model, leaving only three groups and the probabilities of moving between them.
Instead of keeping track of individual students, they worry only about the number of students in each group: Active, Contacted, and Target.
Odds are given to moving from any of these three groups to any of the other two. For example, when an Active member graduates, Griffith assumes for simplicity that a new student enters the college. That is, the number of Actives is reduced and the number of Targets is increased.
The probability connecting Active to Target, then, would be that of a club member graduating.
Griffith’s Numbers. Returning to the club’s poster problem, this model is unable to answer the original “maximum probability” question. However, it provides a way to think about constantly improving the rate of new members—just increase the odds leading into Active!
This is where Griffith and her team hit the wall. Although it might be possible to determine which odds are higher than others—which I did for a club of my own at WKU—, determining the precise value of these numbers is outside both the field of Mathematics and the scope of Griffith’s research course. ∎