Shouldn’t Probability be Vague?
I have always been drawn to probability because of its mysterious qualities. Maybe it is the result of the online poker fad that swept through my high school during the NHL lockout, but the calculation of odds still grasps my attention to this day. What fascinates me the most is how simple rules such as “AND” and “OR” can quickly create a mess of a situation. What begins in high school (or earlier) as a simple fraction that predicts the toss of a coin, soon balloons into factorials, combinations, Pascal’s Triangle, and Probability Density Functions. Despite the complexity of such calculations, they are still only theoretical; anything could still happen. This is a point that I stress to my students whenever we embark on a study of a game of chance.
Quickly back to the toss of a coin. I am assuming this is where most young gaffers get their start. You can imagine my surprise when I first learned that tossing a heads on the first throw does not guarantee a tails on the second, but we still say that the probability of tossing a tails is 1/2. The topic begins to muddy itself very quickly. Then, teachers present multiple tosses in a row. I was floored when I discovered that the sequence HHHHH was just as likely as HTHTT. How could this be? When teachers told me that there was a difference between tossing coins in sequence and throwing a handful of coins in the air, I was a willing skeptic. As if this wasn’t bad enough, during my time as a university student I learned that if I tossed enough coins, any imbalance between heads and tails will certainly happen! (See Stewart, How to Cut a Cake) Probability has its way of taking simple events and creating complex situations. It teaches me that my intuitive calculation of chance may be far off the mark.
I also find that it is this intuitive sense of chance in each student that makes lessons on probability so rich. Students explore the aspects of probability better than any other strand of mathematics. This is a very large statement to make, but probability’s convoluted nature lends itself to intrigue. Not to mention the mystique that pop culture places on risk assessment.
In my junior classes, we play “Homework Roulette”. This is a process (coined by my former teacher Mr. K. Peters) in which a random number is generated and the homework from that section is handed in. It is fairly easy for students to begin to calculate their odds of success, but they are constantly reminded that even if the odds are in your favour, you could be burned.
Probability moves away from this elementary calculation during the weeks of study dedicated to it. Call me crazy, but the challenge of calculating the odds of drawing a certain card from a deck wains quickly. Such problems have a very low floor (they are accessible to many) but a very limited ceiling (there is not much room for students to expand). The reason for this is the parameters with which they are stated. Students reach the end when they devise the nice fraction to represent the risk.
In order to initiate discovery in probability, the situations used must be intentionally vague.
Presenting problems in novel ways creates a battle between the students’ preconceived notion of chance, and the math that dictates otherwise. My favourite example of this comes from a problem I like to call, “Red Card, Blue Card, One Card, Two Card”. I will present the problem, how it is used in my class, and some student reaction.
There are 3 cards. Each is split in half down the middle like a domino. One has both sides painted red, one has both sides painted blue, and one has 1 red side and 1 blue side. If I choose a card at random and show you a side painted red, what is the probability the other side is blue?
The first urge that needs to be fought is revealing the answer. If students expect you to reveal the other side, the question is void. Every time a student asks to see if they are right, I make a very important distinction:
Student – “Mr. Banting, is it blue?”
Me – “You are answering the wrong question. I didn’t ask ‘is the other side blue?’ I asked, ‘What is the PROBABILITY that the other side is blue?’ “
When I begin this class, I always tell the students that I have a quick problem for them. I introduce it with props, and open up the floor for suggestions. I never let students off the hook when they provide a suggestion. Most will say 1/2 and I will ask them where the 1 and the 2 come from. This discussion usually leads to the development of the ideas of options, choices, and favourable choices. What we are doing is developing the ideas of events, sample spaces, and favourable events. I leave out the structure and definitions for now; they will only hamper the discussion. Other suggestions surface. 1/3 is popular. After thinking is explained, I make sure to agree with both.
My non-partisan stance coupled with the vagueness of the problem creates an atmosphere rich with query. Students begin rushing up to the board because they feel that if they only had a marker, the class would sympathize with their logic. Arguments of orientation begin to creep in.
“Let’s say that there are tops and bottoms”
I agree to every ounce of logic that is presented, but leave with a leading question. Are their really tops and bottoms? What if one card is rotated? What if they are on their sides? I pace around as students begin arguing in their desk groups. (I have never seen larger math rages than with this problem) Students grow red in the face as they wait with their hand up to resolve the problem and be the hero.
Soon some hard and fast facts are postured and proven.
“If we see a red side, then it can’t be the blue-blue card”
I agree, name the hypothesis after the student (i.e. the Smith Hypothesis) and then refer to their point periodically throughout the discussion. Students begin to make lists of possibilities and fractions that represent chances. The work is far more valuable because no one has shown them how to do the math. Their intuition leads the way.
I have chosen to end the problem a couple of ways. I have shown them a side of an unchosen card and asked if that changed the problem. Soon options are eliminated from the sample space, and new conclusions and theorems come forward. This is a profitable exercise. My biggest mistake was leaving the 2 unchosen cards unguarded. A student crawled on the floor out of my sight and attempted to rob them right under my nose. Another student actually kicked a desk at me to stop me from leaving before showing the card. Their desire to have it resolved drove them mad. I keep all cards close to me at all times now. I never show them the other side of the card…EVER. Doing this gives them an excuse to say, “I told you so”. It doesn’t allow the haunting to power new learning.
Of course, I could have taught sample space, events, and empirical probability with definitions and textbook questions. Each would have been tightly defined, and the answers would have appeared in the back of the book just as neat. The only problem is that would give the impression that probability is a tidy branch of mathematics. It is definitely not. Having the full-on math argument suits me just fine. The vague problem, creates an open learning atmosphere which emulates the study of probability itself–enigmatic. Until further notice, when it comes to probability in my classroom–vague is in vogue.