Skepticism, Monty Hall, and My Dad
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18-05-2013, 04:49 PM
Skepticism, Monty Hall, and My Dad
I've just come across an interesting correspondence between skepticism, religious stubbornness, and the Monty Hall problem.

The Monty Hall problem, if you're not familiar with it, goes like this:

You're a contestant in a game show, Let's Make a Deal. In this show, the host has offered a choice between three doors. One door, you are told, hides a new car. The other two hide goats (as gag prizes). You get whatever's hiding behind the door you pick. And, because I know there's going to be at least one smartass out there, you want the car, and NOT a goat. You have no indication of which door you should pick, so you pick one arbitrarily. But now things get interesting. Monty opens a different door using the following algorithm: "If you have picked a door hiding a goat, I will reveal the OTHER goat. Otherwise, if you have picked the car, I will reveal whichever goat I choose, or simply pick between the two goats at random." (And yes, Monty was bound by the rules to make this reveal and offer this switch from the start.) At this point, your prize is still unrevealed, but a goat hiding behind one of the other doors HAS been revealed. Monty then offers you the opportunity to switch your choice to the remaining door, before making the final reveal.

Should you switch, and if so why?

For mathematical reasons I will not get into, you should switch. The probability of the car being behind your present choice of door is 1/3, and the probability of it being behind the proffered door is 2/3. This is mathematical fact. It also extremely counterintuitive, to the point where even most mathematicians get tripped up the first time they see it. The natural impulse is to regard the revised probability as 50-50. Coupled with an increased psychological aversion to the possibility of switching away from the correct choice, most people will choose not to switch.

This morning, I mentioned the problem to my dad. He immediately disagreed with the mathematical result (like 90+ percent of the population initially does, and to be honest like I did as well when I first encountered it), claiming that the probability was 50-50, because it was now a choice between 2 alternatives, one good, and one bad. Therein began 5 hours of argument. Watch for parallels with the sort of argument one might enjoy with the religious.

My first approach, appealing to my own authority as a mathematician, didn't carry much weight. Fair enough.

Next, I tried showing him the math. He couldn't follow it, and because he couldn't follow it he wouldn't be persuaded by it.

Then I tried a few simplified metaphors. As such, they were simplistic, and he dismissed them as such.

Next, empiricism. I whipped out my java compiler and programmed a simulated 10000 trials through the problem, with a result of about 66% probability in favor of switching. Dad, a retired programmer, COULD understand the program, at least with a little bit of explanation about what certain functions did. He spent an hour or so staring at the source code, every now and then jumping on little things that I thought were irrelevant. For example, I'd hard-coded that the contestant always chose door #2, and he insisted that the contestant should choose randomly. I expressed that it wouldn't make a difference, but modified the program, and lo and behold it didn't make a difference. He pounced on a potentially-infinite loop and started lecturing on programming style, but when pressed agreed that it wasn't affecting the results we were seeing. (It took a lot of pressing, though.) Finally, he came to the conclusion that the entire logic of the simulation must have a subtle flaw in it, somehow. The fact that he couldn't find the flaw (and he was looking very, very hard for it) didn't mean it wasn't there. And because that would be his conclusion for ANY simulation that didn't support his position, he wouldn't accept a simulation as evidence.

I offered to demonstrate with cards. He declined, saying that face-to-face I'd have too much chance to manipulate his decisions.

Then he turned to Wikipedia's writeup of the problem. Wikipedia agreed that switching had a 2/3 chance of getting the car. However, it also highlighted a lot of early controversy over the problem and how many mathematicians initially got it wrong. He seized on this as further basis for his reservations, while I pointed out how many mathematicians had switched positions and eaten a bit of crow.

He also followed up on Wikipedia claiming this problem was equivalent to to another, and how he couldn't figure out how they were equivalent, and on the basis of this perceived inequivalence held out that the Monty Hall problem couldn't have the same result. He tried phrasing this inequivalence two different ways that I could dismiss to his satisfaction, before settling on a third too vague for me to dismiss outright. I expressed the idea that this was a red herring, and the question of whether or not Monty Hall was equivalent to some other problem wouldn't change the Monty Hall problem. Yet he clung to this inequivalence issue.

Following that, the conversation moved more to the methodological. He was a skeptic, he informed me. It was a learned skepticism -- he comes from a religious background, and only later in his life (before I was born) adopted a more skeptical attitude. I asked him, why then, was he not being more skeptical of his 50-50 stance? Especially in light of the weight of expert opinion, offers of mathematical proof, and even a simulation against it? Why did he insist there were flaws in the simulation he couldn't pin down, or some level of undefinable psychological manipulation by the host which explained it? Where was his skepticism regarding his own position? He protested that I was trying to turn his skepticism back on itself, and I said, hey, that's how skepticism works, you have to question EVERYTHING. Wouldn't that include our own existence, he asked? I explained to him that Descartes, with Cognito Ergo Sum, had done exactly that; not questioning existence, per se, but refusing to rely on it as a fundamental axiom. I asked what sort or degree of evidence would satisfy his skepticism, and he said he didn't know, and that if he could figure that out he'd have a good disproof. I asked what the right level of skepticism was. Where do you stop erring away from avoiding a wrong answer and start erring in favor of accepting a right one? What's the burden of proof?

He said he still had a niggling doubt in the back of his head, and he turned on this feeling that something was WRONG with the 1/3rd vs 2/3rds answer as evidence. He'd intuitively found a flaw, but was as yet unable to express it in words or numbers. Still, he trusted his intuition, thinking that he halfway had a grip on the psychological tendencies that led away from his version of the right answer. Suppose, I countered, that there's something about the right answer which we're just wired to reject, and it's that instinctive but incorrect rejection that you're feeling? He admitted it was possible.

Then he started constructing a narrative in which everyone disagreed with him, but only because they'd been shouted down, and he was the only one pointing out that the emperor had no clothes.

... the whole thing kinda petered out after that.

Replace the "you should switch" answer and the "it doesn't make a difference" answer with atheism and belief, respectively, and the whole exchange sounded really, really familiar. (But, fortunately, there was less invested in the Monty Hall problem, and thus less vitriol in the conversation.)

"If I ignore the alternatives, the only option is God; I ignore them; therefore God." -- The Syllogism of Fail
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18-05-2013, 08:37 PM
RE: Skepticism, Monty Hall, and My Dad
Could your father's reluctance to see your argument as having merit have anything to do with your relationship with him? In other words had a stranger presented the case would he have been more accepting of the results?

I ask this because I get similar pushback from my father and have always wondered if it was the source he was questioning and not the data.

“I suppose our capacity for self-delusion is boundless."
― John Steinbeck, Travels with Charley: In Search of America
“I am quite sure now that often, very often, in matters concerning religion and politics a man's reasoning powers are not above the monkey's." - Mark Twain in Eruption
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18-05-2013, 10:23 PM
RE: Skepticism, Monty Hall, and My Dad
(18-05-2013 08:37 PM)Full Circle Wrote:  Could your father's reluctance to see your argument as having merit have anything to do with your relationship with him? In other words had a stranger presented the case would he have been more accepting of the results?

I ask this because I get similar pushback from my father and have always wondered if it was the source he was questioning and not the data.

Hard to say. Dad always wants to be the Great Pedagogue, explaining everything to everyone, or the Great Jurist, contemplating problems and ruling on them, and he has trouble leaving aside these roles in situations where he isn't actually familiar with a subject. (I think this actually makes it harder for him to become familiar with a subject... and also makes him less likely to WANT to, if he's not already an expert.) He does this with everyone, but I have the uncertain impression that he does it a bit more with me.

"If I ignore the alternatives, the only option is God; I ignore them; therefore God." -- The Syllogism of Fail
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