We humans perform a great many hard cognitive tasks with astounding ease. We form sentences, recognize faces, detect patterns, read body language, and accomplish without effort an astonishing variety of complex feats that turn out to be very, very difficult to program computers to do. This is because our brains have evolved a powerful collection of processing modules whose workings — “under-the-hood” operations that are quite inaccessible to our introspection — have been tuned to a high degree of reliability.
As good as it is, however, this machinery not perfect (how could it be?) and can, when given tasks for which it was never optimized during our evolutionary history, give wrong answers that we are strongly, even irresistibly, inclined to trust.
One area where these weaknesses are evident is the estimation of probabilities, and a fine example is the “Monty Hall problem”.
The setup is simpler enough, and is taken from the old TV show “Let’s Make A Deal”. You are shown three doors, and told that there is a car behind one of them, and goats behind the other two. You get to choose one of the three doors, and you will win whatever is behind it.
Let’s say you choose door #1. At this point the host (that’s Monty) opens one of the other two doors (let’s say door #2) to reveal a goat. He asks you if you would like to stick with your original choice (door #1), or whether you would prefer to switch to door #3.
Now if you are like most people you will think that it really doesn’t matter, from a statistical point of view, whether you switch or not. After all, there are only two unopened doors left, with a goat behind one of them and a car behind the other, so there should be an even chance that you have already got the door with the car. Right?
Wrong. As it turns out, you will win 2/3 of the time if you take the offer to switch, and only 1/3 of the time if you don’t. And if, again, you are like most people, you are going to have a hard time believing this, even after it’s all been explained to you. See for yourself, here.
9 Comments
You’re right that it’s counterintuitive, though I’d assume that one’s probability of winning a prize would go up once it’s revealed that Door 3 is a dud.
My way of looking at it would be like this:
At the outset, when I pick Door 1, the probability (from my perspective) that Door 1 hides a prize is 1/3.
When Door 3 is opened, and I’m offered the choice of switching my choice to Door 2, I’d say that my chances of winning the car are now 1/2, not 2/3. So I’d at least agree that my chances (at least from my non-omniscient perspective) have gone up, but not to 2/3.
Perspective is everything here. It’s a bit like the question of divine omniscience and human freedom. For there to be freedom, something about the future has to be objectively indeterminate (not random or chaotic, since neither randomness nor chaos can be conducive to freedom). If God knows I’m going to order the Big Mac value meal at McDonald’s it’s only a gamble for fellow humans observing my behavior, not for God, who knows the outcome of my supposedly free choice. There are no probabilities from God’s perspective. It’s all written.
By the same token, probability on Let’s Make a Deal is a matter of perspective. The people who placed the prize behind one of the doors are in the godlike position of knowing whether the choice I make will lead to my winning the prize. (But unlike God, they can’t predict which choice I’ll make.)
And that’s why I assume that the probabilities change as my perspective comes to include more information. At first, I have three choices, leading to the 1/3 probability that my choice will be correct. But in the next step, I now have only two choices, which would seem to mean a 1/2 probability of making the right choice.
All of which is to say that I really don’t understand how the probability of winning can reach 2/3 simply by switching choices during that second phase. I’ll be reading the Wikipedia entry to learn more.
Kevin
Hi Kevin,
This has nothing to do with perspective. In the situation described: if you switch, you will win 2/3 of the time. If you don’t, you will win 1/3 of the time.
Apparently the mathematician Paul Erdos was one who refused to believe the answer to this problem until it was demonstrated to him via computer simulation. Also I’m sure you know of the storm of protests Marilyn Vos Savant received, including from dozens of professional mathematicians, when she publicly insisted that switching was the right strategy.
I belatedly see that you linked to Wiki, but another way to see that switching is the correct strategy is to imagine 100 doors, you pick one, and Monty opens 98 other doors to reveal goats. Still want to stay with your original choice?
Malcolm,
Did you ever sense Monty Hall’s presence when you worked at the Power Station?
Ron
My God, Ron, you’re right — I had completely forgotten about that. They used to film Let’s Make A Deal in the building that was Power Station (now Avatar) at 441 W. 53rd Street.
The answer? No. But after all those decades of hit records made in those rooms, you sure sense something when you go in there now.
Hi Dennis,
Well, if even Pal Erdos had trouble with it, then I think everyone else is off the hook.
To my lovely Nina’s infinite credit, the first time I mentioned the conundrum to her she saw the correct analysis right off the bat. “Oh,” she said, “well, the opened door and the other closed door together had a 2/3 chance, but the opened door is no longer a risk, so you’re better off to switch.”
Love that gal.
In every branch of mathematics or science that I have encountered (which is not evenclose to all of them), there is always a paradox; the principles of aeronautical engineering establish that a bumblebee cannot fly. The “problem” lies in the mathematics, flawed because everything scientific is the product of that “flawed” human brain. There is no paradox; the probability of probability is less than 100 percent. Science can make for scintillating conversation, but our “reality” cannot be described by slicing it into parts, “solving” each of the parts, and then reassembling them into the whole. The human brain takes in reality like 35mm film is saturated by the light through the lens, not like a digital camera takes in that light and encapsulates it into pixels onto a drive. If I found myself trying for the car and avoiding the goat, I would follow the “folkloric” principle that applies to multiple choice tests: never change your answer.
Hi, G, and welcome.
I certainly agree with you that we are limited in ways that our limitations themselves make it hard for us even to be aware of.
But we have a much better understanding now of how bumblebees fly, and despite our humble origins and modest abilities, we do continue to make encouraging progress in modeling the world — even if those models must always be approximations.
Alas, if you follow the folkloric principle you mention, and stoutly refuse to waver, you will go home with the goat, two times out of three.