Knowing the Tricks Helps You To Deal With Assumptions

I teach a variety of courses, including one called Puzzle-Based Learning, where we try to teach think and problem-solving techniques through the use of simple puzzles that don’t depend on too much external information. These domain-free problems have most of the characteristics of more complicated problems but you don’t have to be an expert in the specific area of knowledge to attempt them. The other thing that we’ve noticed over time is that a good puzzle is fun to solve, fun to teach and gets passed on to other people – a form of infectious knowledge.

Some of the most challenging areas to try and teach into are those that deal with probability and statistics, as I’ve touched on before in this post. As always, when an area is harder to understand, it actually requires us to teach better but I do draw the line at trying to coerce students into believing me through the power of my mind alone. But there are some very handy ways to show students that their assumptions about the nature of probability (and randomness) so that they are receptive to the idea that their models could need improvement (allowing us to work in that uncertainty) and can also start to understand probability correctly.

We are ferociously good pattern matchers and this means that we have some quite interesting biases in the way that we think about the world, especially when we try to think about random numbers, or random selections of things.

So, please humour me for a moment. I have flipped a coin five times and recorded the outcome here. But I have also made up three other sequences. Look at the four sequences for a moment and pick which one is most likely to be the one I generated at random – don’t think too much, use your gut:

1. Tails Tails Tails Heads Tails
2. Tails Heads Tails Heads Heads
3. Heads Heads Tails Heads Tails
4. Heads Heads Heads Heads Heads

Have you done it?

I’m just going to put a bit more working in here to make sure that you’ve written down your number…

I’ve run this with students and I’ve asked them to produce a sequence by flipping coins then produce a false sequence by making subtle changes to the generated one (turns heads into tails but change a couple along the way). They then write the two together on a board and people have to vote on which one is which. As it turns out, the chances of someone picking the right sequence is about 50/50, but I engineered that by starting from a generated sequence.

This is a fascinating article that looks at the overall behaviour of people. If you ask people to write down a five coin sequence that is random, 78% of them will start with heads. So, chances are, you’ve picked 3 or 4 as you’re starting sequence. When it comes to random sequences, most of us equate random with well-shuffled, and, on the large scale, 30 times as many people would prefer option 3 to option 4. (This is where someone leaps into the comments to say “A-ha” but, it’s ok, we’re talking about overall behavioural trends. Your individual experience and approach may not be the dominant behaviour.)

From a teaching point of view, this is a great way to break up the concepts of random sequences and some inherent notion that such sequences must be disordered. There are 32 different ways of flipping 5 coins in a strict sequence like this and all of them are equally likely. It’s only when we start talking about the likelihood of getting all heads versus not getting all heads that the aggregated event of “at least one head” starts to be more likely.

How can we use this? One way is getting students to write down their sequences and then asking them to stand up, then sit down when your ‘call’ (from a script) goes the other way. If almost everyone is still standing at heads then you’ve illustrated that you know something about how their “randomisers” work. A lot of people (if your class is big enough) should still be standing when the final coin is revealed and this we can address. Why do so many people think about it this way? Are we confusing random with chaotic?

The Law of Small Numbers (Tversky and Kahneman), also mentioned in the post, which is basically that people generalise too much from small samples and they expect small samples to act like big ones. In your head, if the grand pattern over time could be resorted into “heads, tails, heads, tails,…” then small sequences must match that or they just don’t look right. This is an example of the logical fallacy called a “hasty generalisation” but with a mathematical flavour. We are strongly biassed towards the the validity of our experiences, so when we generate a random sequence (or pick a lucky door or win the first time at poker machines) then we generalise from this small sample and can become quite resistant to other discussions of possible outcomes.

If you have really big classes (367 or more) then you can start a discussion on random numbers by asking people what the chances are that any two people in the room share a birthday. Given that there are only 366 possible birthdays, the Pigeonhole principle states that two people must share a birthday as, in a class of 367, there are only 366 birthdays to go around so one must be repeated! (Note for future readers: don’t try this in a class of clones.) There are lots of other, interesting thinking examples in the link to Wikipedia that helps you to frame randomness in a way that your students might be able to understand it better.

10 pigeons into 9 boxes? Someone has a roommate.

I’ve used a lot of techniques before, including the infamous card shouting, but the new approach from the podcast is a nice and novel angle to add some interest to a class where randomness can show up.