I gave my first day lesson on a counter-intuitive problem, and then began using experimental probability to verify our results. I reserved the end of class to remind students what we were doing. I got onto my soap box and began preaching the importance of reality; although we were calculating odds, probability is a risk assessment. Lady luck can be fickle.
As the unit progressed, we saw many uses of probability. We designed our own simulations to specific specifications, and carried out simulations to solve specific problems. Things were going swimmingly from my point of view. I have one student who latched onto probability; he has an uncanny ability for this branch of mathematics. He asked if there was anything he could do to earn extra credit to ensure his mark would be above passing when semester's end came. I had garnered a question from twitter that I was dying to use, but could not budget time for. Until he approached me, I had resigned myself to the fact that I would have to keep it in my bag of tricks until next fall. Sensing an opportunity, I pounced and posed the following problem:
You have 8 balls, and 2 bins. 4 of the balls are Red, and 4 of the balls are White. Your job is to arrange the balls in the two bins however you like, but every ball must be put in one of the bins. (ie. no throwing balls away). I will then choose one of the 2 bins, and then draw a ball from that bin. If I draw a White ball, I win; if I draw a Red ball, you win. Which arrangement gives you the best chance at winning the game?
I cannot remember who gave me this question, and no one seems willing to fess-up online. I can only hope they read this post, and accept my thanks, because what followed was nothing short of amazing. The student set out graphically (as I assumed he would) and soon had a sample space of every possible arrangement. It is important to keep in mind that he had limited training in probability theory up to this point. Basic theoretical and experimental probability was all that was covered. He began to rationalize his way through one trial:
"If one bin is empty, and you choose it, then there is no chance of you winning."
I then reminded him that I must draw a ball. He pleaded with me to allow no balls to count as a loss for me--I agreed. This breakthrough in reasoning was more than enough to convince me. This was as far as he got on day one--a verbal explanation of certain trials.
On day two, he stopped me in the halls with the claim that he could not solve it. I was surprised to hear this based on his great advancements of last day. He said that another student told him that he needed an "N p R" button, and he didn't have one. All of a sudden, he knew there was a rule, and it killed all discovery. I assured him he was on the right track, and he re-doubled his efforts.
The breakthrough occurred on day three. He came in and showed me that he knew certain arrangements would be more likely. Again, he has no knowledge on the additive (OR) or the multiplicative (AND) identities in probability, so this was pure intuition. His calculations were a little off, but the logic was sound. I told him that he needed to work out the chances of every case for the marks, and his response blew me away:
"If this is true probability, Banting, why don't we play the game for the marks?"
Silence. The genius was all there. That was the purpose of the assignment. If he was so wrong in his calculations, then probability should show it. He was essentially challenging me to stop talking the talk, and start walking the walk. I told him that the game sounded like a good idea, and then the bell rang. I went to the staff room and solved the question for myself; I wanted to know what I was up against. Could he choose any combination and just get lucky? After a frustrating (and failed) attempt at calculus, I set out to trials and a Probability Function. I am not going to include the math here--that is for another entry.
The point is, tomorrow I will play this game for bonus marks. The learning is already there to support this decision, and my compliance demonstrates my faith in probability. It may or may not serve as a lesson to all that gambling never pays. To me, this student is using mathematics to its fullest--for human gain. He is manipulating the figures to fall in his favour, even though he has never encountered a hard-and-fast rule. The beauty about his request, is (1) it challenges what I have always preached--that probability is only an assessment of risk, and (2) affirmed what I have always believed--that probability is an excellent vessel for playful discovery.