Have you ever stumbled across a problem that seemed innocent at first glance, but turnedout to be a logical trap? A rabbit hole from which your attention falls deeply, and for aprolonged period of time. That was this problem for me:
Kesha is standing at the edge of a pool, and she’s a little disoriented. Each step she takes, she has a \(\frac{1}{3}\) chance of stepping toward the pool, and a \(\frac{2}{3}\) chance of stepping away from the pool.What is the exact probability that she will eventually fall into the pool? Note that his probability will be more than \(\frac{1}{3}\)- she could enter the pool on the very first step, but there are other ways it can happen.
This is a quote from number 17 of Problem Set 5 of the Educational Development Center's (EDC) Mathematical Immersion for Secondary Teachers (MIST) program from the 2019-2020 school year. (Cuoco & McLeod, 2020) From this prompt, we will explore wide ranging subjects within mathematics. These tangents that we will explore have captivated my imagination, and engaged my thoughts in a way that no other mathematics problem has before.
Students will be posed the isomorphic (but more secondary school appropriate) prompt:
To rephrase the question posed in the video:
A robot is placed one step away from a pool. The Robot is programmed in such a way that it has a \(\frac{1}{3}\) chance of stepping toward the pool, and a \(\frac{2}{3}\) chance of stepping away from the pool. What is the exact probability that it will eventually fall into the pool? Note that his probability will be more than \(\frac{1}{3}\)- the robot could enter the pool on the very first step, but there are other ways it can happen.