A Puzzling Thought

Today I presented one of my favourite puzzles, the Monty Hall problem, to a group of Year 10 high school students. Probability is a very challenging area to teach because we humans seem to be so very, very bad at grasping it intuitively. I’ve written before about Card Shouting, where we appear to train cards to give us better results by yelling at them, and it becomes all too clear that many people have instinctive models of how the world work that are neither robust nor transferable. This wouldn’t be a problem except that:

1. it makes it harder to understand science,
2. the real models become hard to believe because they’re counter-intuttitve, and
3. casinos make a lot of money out of people who don’t understand probability.

Monty Hall is simple. There are three doors and behind one is a great prize. You pick a door but it doesn’t get opened. The host, who knows where the prize is, opens one of the doors that you didn’t pick but the door that he/she opens is always going to be empty. So the host, in full knowledge, opens a known empty door, but it has to be one that you didn’t pick. You then have a choice to switch to the door that you didn’t pick and that hasn’t been opened, or you can stay with your original pick.

Based on a game show, Monty Hall was the name of the presenter.

Now let’s fast forward to the fact that you should always switch because you have a 2/3 chance of getting the prize if you do (no, not 50/50) so switching is the winning strategy. Going into today, what I expected was:

• Initially, most students would want to stay with their original choice, having decided that there was no benefit to switching or that it was a 50/50 deal so it didn’t make any sense.
• At least one student would actively reject the idea.
• With discussion and demonstration, I could get students thinking about this problem in the right way.

The correct mental framework for Monty Hall is essential. What are the chances, with 1 prize behind 3 doors, that you picked the right door initially. It’s 1/3, right? So the chances that you didn’t pick the correct door is 2/3. Now, if you just swapped randomly, there’d be no advantage but this is where you have to understand the problem. There are 2 doors that you didn’t pick and, by elimination, these 2 doors contain the prize 2/3 of the time. The host knows where the prize is so the host will never open a door and show you the prize, the host just removes a worthless door. Now you have two sets of doors – the one you picked (correct 1/3 of the time) and the remaining door from the unpicked pair (correct 2/3 of the time). So, given that there’s only one remaining door to pick in the unpicked pair, by switching you increase your chances of winning from 1/3 to 2/3.

Don’t believe me? Here’s an on-line simulator that you can run (Ignore what it says about Internet Explorer, it tends to run on most things.)

Still don’t believe me? Here’s some Processing code that you can run locally and see the rates converge to the expected results of 1/3 for staying and 2/3 for switching.

This is a challenging and counter-intuitive result, until you actually understand what’s happening, and this clearly illustrates one of those situations where you can ask students to plug numbers into equations for probability but, when you actually ask them to reason mathematically, you suddenly discover that they don’t have the correct mental models to explain what is going on. So how did I approach it?

Well, I used Peer Instruction techniques to get the class to think about the problem and then vote on it. As expected, about 60% of the class were stayers. Then I asked them to discuss this with a switcher and to try and convince each other of the rightness of their actions. Then I asked them to vote again.

No significant change. Dang.

So I wheeled out the on-line simulator to demonstrate it working and to ensure that everyone really understood the problem. Then I showed the Processing simulation showing the numbers converging as expected. Then I pulled out the big guns: the 100 door example. In this case, you select from 100 doors and Monty eliminates 98 (empty) doors that you didn’t choose.

Suddenly, when faced with the 100 doors, many students became switchers. (Not surprising.) I then pointed out that the two problems (3 doors and 100 doors) had reduced to the same problem, except that the remaining doors were the only door left standing from 2 and 99 doors respectively. And, suddenly, on the repeated vote, everyone’s a switcher. (I then ran the code on the 100 door example and had to apologise because the 99% ‘switch’ trace is so close to the top that it’s hard to see.)

Why didn’t the discussion phase change people’s minds? I think it’s because of the group itself, a junior group with very little vocabulary of probability. it would have been hard for the to articulate the reasons for change beyond much ‘gut feeling’ despite the obvious mathematical ability present. So, expecting this, I confirmed that they were understanding the correct problem by showing demonstration and extended simulation, which provided conflicting evidence to their previously held belief. Getting people to think about the 100 door model, which is a quite deliberate manipulation of the fact that 1/100 vs 99/100 is a far more convincing decision factor than 1/3 vs 2/3, allowed them to identify a situation where switching makes sense, validating what I presented in the demonstrations.

In these cases, I like to mull for a while to work out what I have and haven’t learned from this. I believe that the students had a lot of fun in the puzzle section and that most of them got what happened in Monty Hall, but I’d really like to come back to them in a year or two and see what they actually took away from today’s example.