The “Monty Hall Dilemma” is a classic probability problem that stumps most people. Named after the original host of “Let’s Make a Deal,” it begins with three curtains. Behind one curtain is a car, and behind the other two curtains are goats. The participant chooses one curtain, and then one of the two curtains not chosen is lifted, and it always reveals a goat. The participant is then asked whether he or she wants to stay with the original choice or choose the other unlifted curtain. What should the participant do? Does he or she increase the chances of winning the car by switching or should the participant hold on to the original choice? Does it even matter?
The answer is that the participant definitely increases his or her chances of winning by switching the choice. When all three curtains were down, the participant had a 1-in-3 chance of winning the car and a 2-in-3 chance of picking a goat. Thus, the participant is more likely to choose a goat. Once a goat is revealed, however, one wrong choice is removed. If the participant stays with the original curtain, the chance is still 1-in-3. Thus, the curtain that is still down but unpicked must have the remaining probability for the car, which is now 3-in-3 minus 1-in-3, or 2-in-3. So the remaining unpicked curtain is more likely to have the car behind it, and this means the participant should switch. I have explained this to many, many people, and most would not accept the explanation, which is correct. 1
Well, it turns out that one of the people who did not believe my explanation rather sheepishly sent me the reference to a great article demonstrating that pigeons are better at figuring out this dilemma than people.
The article reports on a study done by Walter T. Herbranson and Julia Schroeder.2 In the study, the researchers presented a pigeon with an apparatus that had three keys. The keys would light up to show that bird feed was available “behind” one of the three keys. The pigeon would peck one key, and then one of the two unpecked keys that did not have bird feed would turn off. Now there were only two lit-up keys, and the pigeon had to choose again to have a chance at the bird feed. The pigeon could peck the same key again or the other key that was still lit. This experiment was conducted on several pigeons for many days, and the birds quickly got the right strategy. After thirty days, almost all of them had begun switching the choice after the first key was turned off.
What’s really cool is that twelve undergraduate students were given a similar test, and even after 200 trials, they had failed to adopt the correct strategy. Thus, at least when it comes to probabilistic problems like the Monty Hall Dilemma, pigeons seem to be smarter than people.
What does this have to do with evolution? Well, there are two things to consider. First, the intelligence of chimpanzees is often used as evidence that people and chimps share a common ancestor. Just as similarities in biological structures are supposed to indicate common ancestry, similarities in behavior and intelligence are supposed to indicate the same thing. Of course, as Simon Conway Morris has shown in his book Life’s Solution, we know that’s not really true. There are many instances in which similar structures and similar behaviors cannot be the result of common ancestry. This study simply adds to Morris’s already spectacularly successful catalog of such instances.
The second thing to consider is what the researchers said about why pigeons might be better at this dilemma than people. According to the researchers, pigeons seem to be very empirical. They try things out to see what works, taking their lead from the “data” that they collect. Humans, on the other hand, develop “rules of thumb” that they use to interpret the world around them. Such rules of thumb are nice, but they can often fool people to go against the data.
As a scientist, I cannot imagine how anyone can honestly look at the data and seriously believe in evolution (as meant colloquially – fish becoming amphibians, etc.). The data simply do not support the idea. Of course, since I was trained as an experimentalist, perhaps I think empirically. Perhaps some evolutionists are so locked into their “rule of thumb” of common ancestry that the data aren’t as important to them. Thus, instead of letting the data lead their thinking, they force the data to conform to their “rule of thumb.”
All I can say is that in this case, I am glad that I think like a pigeon!
REFERENCES
1. Donald R. Mack, The Unofficial IEEE Brainbuster Gamebook, p. 76, 1992.
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2. Walter T. Herbranson and Julia Schroeder, “Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma,” Journal of Comparative Psychology 124:1-13, 2010.
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Do you mind explaining the solution to the “Monty Hall Dilemma” one more time? I don’t understand how the situation after one goat is revealed is distinguishable from a situation where one makes the initial choice between a single goat and a car. Basically, it seems that in the “Monty Hall Dilemma” setup, after one curtain is lifted and a goat is revealed, the participant gets to make a fresh choice between two curtains, one concealing a goat and the other a car; thus, the chance that the car is behind a particular curtain should be 50%.
Puzzled, perhaps Andy’s post would make things more clear. He explains it rather well. I quote:
“Here’s the deal with Monty Hall. The probably of the car being behind curtain A is 1/3. The probably of the car being behind curtains B or C is 2/3. Switching lets you get the benefit of effectively picking both curtains and getting the car if it is behind either one.”
Three things….
1. This is me commenting more! (Though it’s a lot easier to comment when you throw in references to pop culture. Who doesn’t love Monty Hall?!)
2. You’re such a scientist! You’re not taking into account that people don’t just think about statistics. I’ll admit I’m still a little confused on why it’s better to switch your choice (must be that ‘rule of thumb’ mentality – please don’t feel the need to explain… I’m good.. really) but regardless, you’re not factoring in the human element. If a pigeon pulled a slot machine lever 100 times and NEVER got bird seed… it would probably stop pulling. But as humans we have that (completely illogical) emotion that tells us, the next pull will be the big one. In your example, I suspect humans would argue they “felt” something about curtain #1 which is why they stuck to their choice. I don’t think that makes pigeons smarter than us, just more logical perhaps.
3. Way to take something fun the ‘Let’s Make a Deal’ (perhaps one of the stupidest, yet most addicting shows to ever air) and turn it into a boring “science” thing! And you wonder why I call you geeky?!?!
Black Sheep,
1. WOOT!!!!
2. Good point. Basically, if we as people WANT something bad enough, all the data in the world won’t change our mind. If you could possibly stomach learning something you aren’t really interested in 🙂 , you might read Andy’s comment.
3. You forgot Nerdy!
What if there were three curtains, you chose one and a none goat one was raised at that time you have a 50% chance of getting it right if you switched. But what if you had chosen the other curtain that was not raised instead? Your chance would still be 50% right? I agree with your explanation and I see logic in switching because it would improve your chances from 1/3 to 1/2. I do not know if you would get a car more or less often though.
John, you aren’t thinking correctly about the problem. You don’t change your probability to 1/2. You change it to 2/3. There is a 2/3 chance that the curtain you DIDN’T choose has the car. Since Monty will always reveal a goat, the 2/3 chance is now left to the other curtain – the one you didn’t choose.
If Monty didn’t ALWAYS reveal a goat, then the odds are different, but Monty will always reveal a goat. That’s what makes this so tricky to think about.
Jay, I agree, you certainly are a bird brain! (humor)
“Perhaps some creationists are so locked into their “rule of thumb” of a young earth that the data aren’t as important to them. Thus, instead of letting the data lead their thinking, they force the data to conform to their “rule of thumb.””
As with most of your arguments, which are simply opinions strongly stated, it can be reversed to attack your position. Your “rule of thumb” – the earth is not older than 10,000 years. Why don’t you let the data lead you to the truth?
Shooter, in this case, I am pleased to be a bird brain, because it means I am allowing the data to guide my views.
I certainly agree that the statement can be reversed, but it cannot be done so meaningfully. The data speak strongly in favor of a young earth. Thus, if I had a “rule of thumb” for a young earth, it would simply be confirmed by the data.
The other problem with your statement is that I don’t have a “rule of thumb” for a young earth. Indeed, at one time when I didn’t know much science, I thought the earth was as old as my teachers and textbooks told me. The more science I learned, however, the more it didn’t make any sense. If anything, then, my “rule of thumb” was that the earth was 4.6 billion years old. An open-minded look at the data changed that, however. So the data did lead me to the truth. Why do you continue to ignore the data?
Ben makes a great point, Shooter. Nearly everything you post is demonstrably false. Often you tenaciously hold on to your false statements for a while until the data overwhelm you and you are forced to give up and either change the subject or try to mock me or some other serious scientist. Thus, your track record when it comes to data is pretty abysmal.
On the other hand, your track record when it comes to fervent faith in the high priests of science is very strong…
I got the point about Monty Hall but not the pigeons.
Here’s the deal with Monty Hall. The probably of the car being behind curtain A is 1/3. The probably of the car being behind curtains B or C is 2/3. Switching lets you get the benefit of effectively picking both curtains and getting the car if it is behind either one.
Shooter, what do you know about data? I’m certain you know about the orthodox interpretations of data, but what of the data itself? Earlier you talked about consensus of scientists, but why should a form of priesthood of scientism interpret the data for anyone? Because they are specialized? Dr Wile is specialized in Nuclear Chemistry. Dr Don Batton is specialized in Plant Physiology. Dr Jonathan Sarfati is specialized in Physical Chemistry. Et al. They, qualitatively, would be able to interpret within their fields as well as understand the general concepts pertaining to scientific inquiry. But even apart from them or any other Ph.D holding scientist, what do you actually know about data or do you primarily know just the orthodox doctrinally acceptable interpretations of data taught in government operated quasi educational facilities?
I don’t want to waste anyone’s time on this rather trivial issue of the “Monty Hall Dilemma”, but let me come at it once more.
As I said before, after one goat is revealed, it seems that, in effect, the participant gets to make a fresh choice: either choose the remaining unselected curtain, or re-choose the initially-chosen curtain. Since there are only two possibilities, one car and one goat, the probability of choosing the car should be 1/2.
It seems fallacious to treat the probability that the initial choice was correct as remaining at 1/3 after one option is eliminated. The probability changes since the number of options changes.
For example, if I blindly select a marble from a sock that contains two white marbles and one black marble, the probability of picking the black is 1/3. However, if my first pick is a white marble, the probability of picking black on my second try is 1/2. My subjective desire to select the black marble does not mean the probability of success remains the same after one option is eliminated.
Puzzled,
Here are the possibilities:
A B C
——-
c g g (case 1)
g c g (case 2)
g g c (case 3)
Let’s say you pick curtain 1 and don’t switch. As you can see, the car in behind curtain A only in case 1, which is 1/3 of the time.
Let’s say you pick curtain 1 and switch. In case 1, you don’t get the car, in case 2 and case 3 you do get the car — so 2/3 of the time you win the car.
Andy, thanks for the excellent explanations! The point about the pigeons is that they figured it out by observation. The undergrads couldn’t do that. They let their preconceived notions trump the data. The pigeons let the data guide their actions.
Puzzled Reader, try to imagine it from a different angle. After you select one of the curtains, you know that at least one of the two curtains that you picked does not have the car behind it. Now suppose that instead of showing you which one didn’t have the car Monty just let you decide either to keep your original pick or select both of the other two curtains (if either of the other two curtains has the car, you get it). Just because you are shown which one of the two doesn’t have the car doesn’t make any difference to the probability
Your marble idea is not quite the same thing. If you randomly selected one of the 3 marbles out of your pocket you would have 1/3 chance of getting black 2/3 chance that the black stayed in your pocket. But what if, after you had selected a marble, a white marble that was left in your pocket disappeared, would you be more or less likely to have a black marble in your hand or in your pocket. You would be more likely to have the black marble in your pocket.
Nice explanation Andy, I didn’t see your comment until after I had already commented.
Andy,
Kudos for explaining something more clearly than the dear Dr. In my experience, that doesn’t happen often.
Isn’t there more to it though? For the math to stick, don’t you have to make the assumption that each case is used equally? Or at least that cases 2 and 3 collectively are used more frequently than case 1? For example, let’s say I play the game 10 time and pick A every time. Because I’m a human and not a pigeon, I refuse to switch because I “feel” A is the curtain with the car. Well, if Monty sets the game up using case A 6 times and cases 2 and 3, four time collectively… I get 6 cars! But if I had switched each time, I’d only have 4. Right?
Dr. Wile… see I can pretend to interested!
Black Sheep and Puzzled, remember that Monty will always choose a goat. If Monty also chose at random, then 1/3 of the time, he would reveal the car, and you would lose right away. That would “suck up” 1/3 of the wins, which changes the strategy completely.
However, that’s not how the game is played. Monty always chooses a goat. Thus, as Andy said, switching your choice allows you to actually choose two curtains: Monty’s (which always has a goat) and the other one. That gives you a 2/3 chance of winning.
And thanks for pretending, Black Sheep!
Jay, first, have a good weekend. Second, I would love to look at data that make a positive case for the earth being between 6,000 and 10,000 years old. By positive case, I mean to exclude the various data that you misinterpret to show that the earth can’t be older than 10,000 years old. I mean what is the data that makes the case the earth is 8,000 years old +/- 2,000 years. What is this data?
Shooter, you have already seen the data. They are shown in the “age of the earth” category. You can CLAIM the data are misinterpreted all you want, but that doesn’t make it so. The data are interpreted correctly, and they indicate a young earth. They also don’t require as much unscientific extrapolation as is required to believe in an ancient earth. Finally, two of the dating methods (the earth’s magnetic field and the helium in zircons) made predictions about data that had not been measured, and once the data were measured, the predictions were confirmed. That’s EXACTLY how positive scientific evidence works. Thus, a young earth is the more scientific position to take. You can choose not to take that position, but in my opinion, you are making that choice against the data.
Honestly, I wouldn’t care if you made the choice of an old earth based on the data. There are scientists I respect and admire who believe in an ancient earth, because they can make reasonable arguments for why the ancient-earth-supporting data are more important to them. I disagree with them, but I can at least see their point. The problem is that you have not really looked at the data, as is made clear by your comments. Instead, you have made your choice because of your fervent faith in the statements made by the high priests of science. While you can choose to believe that way if you wish, don’t claim that you are being rational about it. You clearly are not. Also, don’t try to claim that others who have actually studied the data are misinterpreting data you haven’t even bothered to interpret yourself.
By the way, the word “data” is plural. Thus, your question should read, “What ARE these data?”
Thanks for catching the italics thing!
oh, and you didn’t close an italics tag in your citations. That’s why all the comment and sidebar text is italicized.
double oh, my gravatar is Ole Einar Bjorndalen, the best biathlete of all time. (This is a test: Bj%C3%B8rndalen)
To all the commenters, rejoice, and be glad!
Black Sheep,
If we assume a random distribution so that each time the car is likely to be behind any particular curtain 1/3 of the time, then the math works out as I have suggested. The situation that you proposed does not meet that criteria, and so that is why it doesn’t work.
Andy, I have to agree with the first part of Black Sheep’s comment. You certainly did explain the situation MUCH better than I did. I am not sure such a thing is as rare as Black Sheep thinks it is, but you definitely have a knack when it comes to explaining this kind of math!
Thanks, guys, for the kinds words. I like this problem because it is so counter-intuitive on the surface. Several years ago I worked as a crypto-math guy for the DoD and we had many side discussions about Monty Hall and other math-type puzzles. This blog post brought back many fond memories from those days.
Andy, thanks again for your excellent explanations. I hope you lend your wisdom on more posts!
So basically there is a 2 in 3 chance that your picked curtain holds a goat, in which case once the curtain and the shown goat are removed, only a car remains? Very clever.
Better strategy–listen for the bleating.