Ben Hoban is a GP in Exeter.
I didn’t really get the Monty Hall problem first time around either. To start with, it seems obvious; then, when you think about it, you can see that there’s something else going on, although it can take a while to put your finger on it. In the end, it turns out that it isn’t all about the doors, but I’m getting ahead of myself now. Just to make the point that this isn’t a straightforward sort of thing, and despite its name, the problem actually has nothing to do with Monty Hall, former host of the US game-show Let’s Make a Deal. It was originally posed in 1975 by Steve Selvin in a letter to The Statistician and became notorious after Marilyn vos Savant published the solution in Parade in 1990, and as far as I know, neither one of them is or ever was a game-show host.1
The fact of the matter is that the set-up of the problem only really works if you can imagine it on the set of a cheesy game-show.
The fact of the matter is that the set-up of the problem only really works if you can imagine it on the set of a cheesy game-show. Three doors are visible to the audience, behind one of which is a car; behind the others are two goats, presumably sedated to stop them giving themselves away. I should make it clear that within the terms of the problem, cars are good and goats are bad, no offence to goats. Anyway, a contestant has to choose one of the three doors, and when it’s opened, they get whatever’s behind it. First, though, the game-show host, who can see what’s behind each door, opens one of the remaining two, revealing a goat, and offers the contestant the chance to change their mind. The question is: should they stick with the door they originally chose, or go instead for the other unopened one?
The obvious answer is that if there are two doors left, then one of them will be hiding a goat and the other, a car. In this case, the odds of ending up with either must be 50:50, and it doesn’t matter which of the doors the contestant opens. This would certainly be true if we only had to consider those two closed doors, but what difference does it make that we started with three doors and two goats, and that the host deliberately opened a door with a goat behind it? You can play a simplified version of the game at home with a volunteer in the role of contestant, for example by using three upturned cups with a sweet hidden under one of them. If you play the game often enough, it soon becomes clear that you’re twice as likely to win if you change your mind, although you may also find yourself explaining Monty Hall to your dentist.
The reason it pays to change isn’t complicated, although it is counter-intuitive. When the contestant first chooses a door, they have a 1/3 chance of winning the car, and a 2/3 chance of losing (or winning a goat, depending on your point of view). When the host opens one of the remaining doors, these probabilities don’t change: the contestant is still twice as likely to have picked a losing door as the winning one, although now there is only one alternative rather than two. You should always change, and this is the answer that vos Savant gave.2
This may not seem terribly interesting, although it caused the 1990s equivalent of a social media pile-on, with huge numbers of “experts” writing in to correct what they saw as not just bad maths on vos Savant’s part, but bad form. They were confident that the odds were 50:50, and offended that rather than bowing to their expertise, she stuck cheerfully to her amateur guns. For those unfamiliar with American periodicals, Parade magazine is all about entertainment, recipes, health, life and holidays: perhaps not the first place you’d look for the answer to a probability puzzle. Vos Savant’s regular spot was a column called Ask Marilyn, in which she tackled reader questions on a broad range of subjects. Her main qualification was the fact that she had the highest IQ ever recorded, but perhaps that wasn’t enough to make the experts listen to her; the good news is that they got there in the end.3
The funny thing about the Monty Hall problem is that it’s really a human story dressed up as a probability puzzle, and the second goat isn’t there to distract us from an obvious truth, but to help us understand a bigger one. The day-to-day realities of general practice encourage us to offer people simple solutions to complex problems: a pill, a test, a referral. We are short on time and resources, and beleaguered and brow-beaten by politicians, news media, and sometimes our patients, who worry not just about their symptoms, but about being ignored or falsely reassured, and there is a constant drive to do something. It would be perverse to look down our nose at a solution just because it’s simple, but also dangerous to take it at face value when most things in life are more complicated, and when routinely preferring simple solutions makes it harder to think of others that might work better.
It would be perverse to look down our nose at a solution just because it’s simple, but also dangerous to take it at face value when most things in life are more complicated…
In any consultation, then, we’re in a similar position to the contestant, having to choose well with limited information. In our case, we need to understand not just a patient’s symptoms, but also the story that places them in their proper context and tells us what problem we’re actually trying to solve. We are like the game-show host too, using inside knowledge to guide the contestant: the key for us is to show someone that we’re prepared to see things their way first, so that our biomedical perspective helps us make a good shared decision rather than crowding out their concerns. As generalists providing personal care, we also resemble the good-natured magazine columnist, to whom people are happy to bring their questions, even though they sometimes doubt whether she can answer them. Our challenge here is to build positive relationships with people who may be ambivalent about our role in the health service and have been forced by circumstances to depend on someone they haven’t yet learned to trust.
If doctors are also a kind of expert, our expertise certainly isn’t the kind that means we’re always right, or even necessarily better-informed than our patients, but it still allows us to do something, even when there aren’t any simple solutions. Whether we’re trying to understand, advise, or build trust, the one thing we can always do is to listen properly to what someone is telling us.4 The answer may seem obvious at first; then, gradually, we might start to put our finger on what’s really going on; in the end, the question could turn out not to be different to what we thought it was. There will always be times when we pick a goat, because that’s the nature of the game, but we can still do our best to play it well.
References
1. Selvin, Steve (February 1975a). “A problem in probability (letter to the editor)”. The American Statistician. 29 (1): 67–71. doi:10.1080/00031305.1975.10479121
2. vos Savant, Marilyn (9 September 1990a). “Ask Marilyn”. Parade. p. 16. Archived from the original on 21 January 2013 https://web.archive.org/web/20130121183432/ http:/marilynvossavant.com/game-show-problem/
3. Leonard Mlodinow, The Drunkard’s Walk: How Randomness Rules Our Lives, Penguin, 2009
4. Rita Charon, Narrative Medicine: Honoring the Stories of Illness, Oxford University Press, 2008
Featured photo by Naser Tamimi on Unsplash
