Posted on 06. Jul, 2007 by The Gimcracker in Science
I have always been fascinated with counterintuitive puzzles. During my third year in quantitative analysis at IUPUI I was presented with the famous Monty Hall problem. It states:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Another way to say it in less words is:
In search of a new car, the player picks door 1. The game host then opens door 3 to reveal a goat and offers to let the player pick door 2 instead of door 1. Should the player switch?
Would you believe that you actually double your chances if you switch doors, even though the objects behind the doors haven’t changed? Counterintuitive indeed. You’re saying “no way, I’d have to be an idiot to believe that.”
I have never been able to explain this paradox to people without getting into a bunch of details and diagrams and examples and such, which has been pretty frustrating because your average “non-geek” person will just lose interest as soon as they see something shiny. (If detail is what you’re after, check out the all-knowing Wikipedia article on the Monty Hall problem.)
The point of this post is that I finally found a simple way of understanding this logic. Just increase the number of doors. If instead of choosing 1 out of 3 doors you chose 1 out of 100 doors with goats behind 99 of them, and the game host removed 98 doors with goats behind them leaving just your original door and one other door, switching would increase your chances from 1/100 to 99/100. This proves that switching really can increase your odds, even if nothing behind the doors is actually changed. It shows that the game host’s prior knowledge of the location of the prize plays a big part of the original odds.
If that rattled your brain, this will blow your mind:
If you were to fully develop the entire tree for all possible chess moves in a single chess game, the total number of board positions far exceeds the number of atoms in the universe.
Oh, so you disagree with that? Just like you disagreed with the Monty Hall problem? I’ll explain it later because I think I’ve crossed the threshold of geekness for one blog post. I mean, who spends their Friday posting about probabilities and statistics, honestly?
Here’s some pictures of geeks so you can at least get some pleasure out of reading this insanely boring post.
I’m going to be obstinant and say that they are wrong.
When you make your choice the first time, you have a 1/3 chance of making the right choice.
When you make the choice the second time, whether you switch doors or not, you have a 1/2 chance NO MATTER WHAT.
Because you are making a completely new decision in no way related to the first decision. You are just happening to chose that same 1 out of 2 doors again. You still choose though. You are re-choosing no matter what, so in your re-choice you are just happening to choose the same door, and that door has a 50% chance of having the car, whereas before it had a 33% chance. And yeah, I’ve seen the wikipedia page, and they are wrong because (imo) their logic is flawed.
I guess the only way to prove or disprove this would be to write a computer simulation that would stick with it’s door 1 trillion times, and would switch doors 1 trillion times and then see which way came up with more wins. F*** it, that’s what I’m doing for the rest of the day. I’ll post my results when I have them, plus the source code (in PHP since it’s the least common denominator).
I’m taking bets. I’ve got 1 beer (that’s medium quality beer, meaning a six pack of them would cost at least $7, like Harp, Guinness, or Bass) on it not making a significant difference either way (let’s say, the models will be within 7% of eachother).
I’m just going to stare at these nerds until Chris gets back with an answer.
Ok… I’m glad I didn’t say something like “I’ll drink stroh’s out of one of Phil’s boots” or something… because I would be chugging beer from a boot next week.
Brian, even though you never took me up on the bet, even when I IM’ed you after I knew I was wrong to see if you wanted to, I am so in awe of your vastly superior mind at having been able to figure this out without building a working model that I will be giving you a Harp Beer next men’s group.
To see for yourself, go here:
Here is the reason why this is so hard to believe (or at least what was throwing me off):
If you randomly choose whether or not to switch, you have a 50% chance of getting the right door (which you can see proven in my model at the link above).
So, without a method you will have a 50% chance of getting the right door. If your method is to always stick with your door, you have a %30 chance of getting the right door. If your method is to always switch your door, you have a 65% chance of getting the right door.
This proves a point about life. If you are going to have a method, make sure it is founded on scientific knowledge and fact, otherwise you are just better off going with your gut.
Gee… If only I had thought to click one of the links at the end of the article…
oh well, that’s an hour and a beer down the drain…
An hour and a beer are a small price to pay for true understanding. I can’t wait to taste my delicious Harp.
I can’t get enough of that third geek picture. Dude looks like Milton from Office Space mixed with Ace Rothstein from Casino mixed with an owl, and the best part is he’s smoking a cigarette. Is he the precursor to Samuel L. Jackson’s character in Jurassic Park?
I also like counterintuitive puzzles, but the only one that I am familiar with are the Monty Hall 3-Door Problem, and the similar Prisoner’s dilemma. I was hoping that you could point me in the right direction of finding more counterintuitive puzzles of a math,physics, and/or spatial kind. I have been looking for two months now, and aside from the two aforementioned ones, nada. Thanks
Since I am most interested in statistical and probability paradoxes I will do some research and see what I can come up with.
Off the top of my head I can think of the Benford paradox, or the first-digit paradox I think it’s also called. It states that the number 1 is much more likely show up as the first digit in datasets, number 2 is less likely, number 3 even less likely, on down to number 9 which is the least likely to show up as the first digit. It’s very counterintuitive because naturally we think that it should be an even distribution, a 1/9 chance that each digit should show up first (we’re talking random datasets here, like a dataset containing the heights of buildings for example).
The coolest thing about this law is its use in insurance fraud, accounting fraud, and any other instance where someone could make a fraudulent dataset. Most people tend to evenly distribute numbers when they’re coming up with false data, but if you analyze their numbers using Benford’s law you can prove that the numbers where falsely created. According to the law 1 should show up about 33% of the time and 9 should only show up 5% of the time, whereas their data most likely has both 1 and 9 showing up around 10% of the time.
I’m not an expert on this law and it’s been a long time since I’ve studied it, so I could be way off on some of this. You can find out more here.
I’ll post more counterintuitive puzzles as I come across them.
Here’s a few simpler paradoxes similar to the Monty Hall puzzle:
The Necktie Paradox – A situation in which it is actually in both men’s interests to wager their neckties. It seems like it would be 50/50, or favored for one of the men, but counterintuitively it is in fact in both of their interests.
The Sleeping Beauty problem – This one is a bit too long to explain, but it’s really not that complicated. Check out the link to see what I mean.
You can find a good list of paradoxes here. I haven’t had a chance to research all of them, but the ones I’ve looked at are pretty cool.
If I lost 100 pounds, I would look like the second geek.
Here’s a paradox for you:
Proteins are made up of folded chains of amino acids.
There can be any one of 20 different amino acids at any given position.
The arrangement of the amino acids makes all the difference as to the successful folding and final function of the protein.
Only a limited number of arrangements will do anything meaningful (much like a limited number of arrangments of letters will spell a sentence).
The possible combinations of amino acids in just a short protein is trillions of times larger than the number of atoms in the universe.
The simplest known life in the history of the earth consists of hundreds of proteins.
Proteins will not self-assemble — at least not into long (polypeptide) chains.
Ribosomes are required to assemble proteins.
Ribosomes need messenger RNA to tell them what to assemble.
Messenger RNA gets its instruction from DNA.
Numerous other helper proteins and molecules are required to support the DNA –> RNA –> Ribosome –> Protein process.
How does raw chemistry randomly and incrementally assemble into such a complex cellular system without intelligent design?
You know what the most amazing part is? There are a lot of people that will answer you, even after ALL that, with “coincidence”. Though, in my opinion, you have done all but prove that impossible.
Pure atheistic materialism is a stubborn wench though she cannot answer any of the most urgent and passionate questions you pose to her.
I didn’t realize that you believe in intelligent design. I do also, but always felt like it was something I had to keep under wraps, because of the college I attended (UC Berkeley). I once had a professor that set up an experiment to show just how this could happen randomly given enough time. I stated at the time what I thought was obvious “but the experiment wasn’t random, you designed it”. The class laughed–not necessarily out of ridicule. He just wrote off my remark. Stubborn wench indeed.
Hehe, that’s a great retort. I wish I would’ve thought of that. It really points to the fact that statistics, probabilities, and the like are a result of intelligent design, just like your statement to your professor points out.
I read through that DNA –> RNA –> Ribosome –> Protein process again and was almost able to comprehend the sheer magnitude such a coincidence. Almost.
If you want a more detailed (but still understandable) exposition of some of the problems, I invite you to read one of the essays in my Science section. I’d particularly recommend either the 3-part series on “Abiogenesis: A Problem of Origins” or on “Evolution’s Credibility Problem.”
Regarding Nathan’s experience, chances are that the experiment was the famous Miller-Urey one where some amino acids were created (among other useless and detrimental stuff), which are the building blocks for proteins. This experiment has numerous problems, but at best it could only be like saying, “We’ve made bricks! Now we can be confident that mansions can be randomly assembled.”