There’s a major problem with probability: it doesn’t work very well. For example, common knowledge tells me that in a fair coin toss I have 50% odds of flipping heads and 50% odds of flipping tails. But flip a coin once and you’ll either get 100% tails or 100% heads.
Now, I get it. I mean, don’t worry. After many flips, blah blah blah. We can study a fair coin toss for two weeks, counting total possibilities, various games, expected values, special orders with surprising probabilities, etc, but none of this helps me guess the next flip at all. Even in the next six flips there’s a depressingly low chance of getting three heads and three tails. We certainly can’t rely on it happening in our math class while we try to convince people of what “50% likely” means!
I believe in probability, but it is indemonstrable. Once you do something, it 100% happened despite its 50% (or 5%) odds. The challenge for us, then, is to reconcile these seemingly-disparate pieces of evidence.
I chose to let the students have a 30-minute discussion about the issue and I brought several props. My goal was to get the students to seriously consider their beliefs about probability – to be wary before they were confident. By the end of this conversation, I want some students to doubt that heads and tails are equally likely.
I started the conversation by showing them this quarter, through which I have inserted a piece of paper:
After the requisite jokes about how hard it was to get that piece of paper through the quarter, I asked, “What are the odds of flipping tails?”
“50%,” one student immediately asserted. The other students all agreed (I was surprised that they all agreed). Experimentally, we showed that the odds are not even – there was a strong correlation between the side that starts up on your thumb and the side that lands up on the table, apparently because the paper drastically reduced the coin’s tendency to flip at all. It sort of parachuted down instead of flipping and rolling. One of the students tried cutting the paper (which started as a clean square) to see if she could affect the odds. The students’ assumptions were incomplete and wrong, which is one of my favorite ways of getting them interested.
Then I asked another set of questions, this time with a plain coin and no paper:
“When I hold this coin way above the table with tails facing up and drop it, what are the odds that tails will land up?”
50%, Riley, [obviously | I think | assuming it bounces].
“Ok… when I hold this coin one inch above the table with tails facing up, what are the odds that tails will land up?”
100%, Riley, it won’t even flip.
But at two inches, the coin bounced a bit, and at three inches it was hard to tell what was going to happen. “At what height does it become 50%, class?”
They got my point. If we can tell what is going to happen after an inch, why can’t we tell what is going to happen in a foot? Philosophical questions abound here, right? And they started to doubt themselves about everything. They started to see that there’s no way to test the probability with perfect accuracy. They started to see that you might be able to rig a coin flip to turn out the same every time. They started to see the effects of human bias in selection games (I had a deck of cards too).
By the end of this discussion, my students were using phrases like “assuming that there is enough complexity” and “assuming that no one is cheating” and “assuming that the sides are equally heavy.” They saw the assumptions that go into a statement like “heads and tails are equally likely,” and saw that we can’t rely on that information in all cases. I sacrificed 30 minutes of class during which they could have practiced permutations and combinations, but I hope that their increased savvy about statistics will pay off.