Why the Three Doors is Not Such a Difficult Problem

AL GREEN - TAKE ME TO THE RIVER

I posted an article yesterday entitled, "Critical Thinking and the Three Doors." I used a famous logic question as an example that I learned about examining problems. Trust, folks, that I am not mathematics whiz. Far from it. But one good reader questioned my method. They wrote: 


This is one of the world's most famously difficult logical problems, and many of the world's top mathematicians got it wrong at first. If you're going to test people's critical thinking skills, may I suggest that it may be be better not to choose a problem that has stumped Nobel-Prize winners and the world's best mathematicians? 


My only retort is to say that, "Yes, this might be one of the world's most famous logical problems but the fact that it stumped some of the world's top mathematicians and Nobel prize winners shows the extent of the poor critical analysis problem today and that even that these 'highly educated' people having poor problem solving abilities and highlights my original thesis in that "85% of all adults today show poor critical reading and analytical reasoning skills."


Like most, I got this wrong at first. But quickly understood it when I diagrammed it out. It was easily understood when it was first explained to me, because I was taught to diagram problems on paper first by my mother when I was a kid. Even though it is counter-intuitive, these things can take practice but when diagrammed are quite simple.


Try it yourself. Diagram out the three possible outcomes. 


Here's why, this is not a difficult problem and how diagramming things out on paper can help you to "see" what your intuitive mind misses.


Here's a graph showing you all possible outcomes of this game show problem. It is very simple and any 5th grader could have figured this out. The top graph , "Switch from Door 1"shows the possible outcomes if you picked door #1 and switched. You can see that, with all possible outcomes, you get a higher odds of winning if you switch. The bottom graph shows "Stay on Door 1." Staying show, with all possible outcomes, that you will have a 2/3 chance of losing if you stay. 


Like I said, this is simple and any diagramming of this problem would show you the answer. You choose door #1 in every example. In the top, you switch. In the bottom, you stay.

That so many of the world's top mathematicians can get such a simple problem wrong shows the sorry shape of our institutions today. Any elementary school child could graph this out on paper and see what the results are. 


This is also why the "America's top economists" were and are often wrong about the economic situation today. But that is another calamity for another day.


Don't believe that writing things down on paper is not a very simple way to solve problems? Well, it is, and I can prove it to you again. It's not difficult math. The try this one. The old boy, boat, chicken, dog and bag of seeds riddle.


It goes like this; You are a small boy. You have a tiny boat. You are on one side of a swift river. You need to cross the river but your boat is too small for more than you and one other item (the dog, chicken or bag of seeds).


If you take the dog first, the chicken will eat the seeds. If you take the seeds first, the dog will eat the chicken.


If you take the chicken first, fine, but then what happens when you bring over the dog (or seeds) and the have to return to pick up the other? No matter what, under this scenario, something must be left alone with the other.


Oh, and by the way, there are piranhas in the water so you cannot swim along side the boat. 


What do you do? 


I'll let you answer this by yourselves. I can say that if you pull out a piece of paper and do not trust intuition, the answer is simple. The problem is that too many people - even Nobel Peace Prize Winners - do not diagram things on paper and their intuition is completely wrong.


Some recent examples of Nobel Prize winners who are completely wrong might be Al Gore or Barack Obama. Need I say more?

Nowadays, Nobel Peace prizes are given out to people who are completely wrong

Critical Thinking Quiz & The Three Doors

UPDATE AT BOTTOM


The other day, I wrote a blog post criticizing many in the adult population today because of their inability to read critically and exercise basic analytical reasoning. 

WEIRD AL YANKOVIC - I LOST ON JEOPARDY

The title of that blog was Critical and Analytical Thinking are Lost Arts Amongst Many of Today's Adult Population. I wrote:


Most unfortunately, I'm getting the impression that, judging from the comments I often hear and from reading comments on Social Media websites, at least 85% of the adult population are incapable of well-executed critical analysis and reading. We live in a society today whereby far too many people have lost the ability to perform even the most basic levels of critical/analytical thinking when it comes to what they see or hear on TV or what they read in print. This is a very sad situation. People are not able to extract facts from conjecture. They seem to be incapable of deeply considering the motivations of the people they see/hear or read on the mass media.
Proper critical analysis will always examine a person's possible motivations for what they say or write. Too few adults today seem able to exercise this most basic ability.  


I then went on in that post to give several examples of how this works (or doesn't work) in real-life situations. I especially thought it was important because of the recent Fukushima disaster and how many people panicked and fled Japan without thinking carefully through the problem before making any decisions on what to do. It had much to do with how people filter the information that we receive daily from the mass media and how people today seem to be unable to rectify numbers and their relation to what they mean to our lives.


Of course, again, I got many comments criticizing me and even some people wrote nonsense and humorous comments like, "I give your reasoning an 'F' for failure." Most of those comments were deleted because they were so poorly written or were just outside of commenting guidelines. Those sorts of comments are, I believe, mostly from the people who are believers of AGW or think that dolphin slaughter in Japan is a crime... These are people who are definitely blinded by dogma and cannot fairly see the forest for the trees. (I will write separately on that too later).


I did get some mail (to my private mail account too) that were supportive. Thanks.


Now, I'd like to do a very simple logic experiment with you, dear reader. The fact of the matter is that critical reading skills and analytical reasoning are all based on mathematics (thinking with the left-side hemisphere of the brain is for logic, reasoning and math). These mathematics are not difficult. They are, though, in many cases counter-intuitive.


Let's take this short one question quiz and see how you do.

You are a contestant at a game show. There are three doors in front of you. The doors are marked doors, "A", "B" and "C". Behind one of these doors is $1 million USD in cash. Behind the other two doors are ham sandwiches.


You are asked to pick one door. For the sake of this quiz, let's say that you choose door "A". The game show hosts then shows you that behind door "B" is a ham sandwich. He then asks you if you want to keep door "A" or switch to door "C".


Question: What should you do? Should you stick with door "A" or switch to door "C"? 


(scroll down for answer)
...
...
...
...
...
Answer: You should now switch to door "C" as door "C" now has a 2/3 chance of winning. Staying on door "A" is a bad choice because door "A" only has a 1/3 chance of winning.


Just because the host has shown you that behind door "B" is a ham sandwich, that doesn't make your original choice of door "A" a 50/50 chance of winning. 


It is unmistakable that switching will increase your odds of winning from 1/3 to 2/3.


Now, I expect that many will comment (I hope you do) so that I can help you to see that switching is the correct answer and guarantees a 2/3 chance of winning.... Think about it, in old movies guys would do the peanut under the cup gambling with suckers? How do you think they always won? They don't have to cheat.


UPDATE :


Here's a typical example of how the recent Fukushima nuclear disaster is used to spread fear by use of illogical reasoning. If anyone is bothered to investigate or think about it, they'd see that the chart below says many things:

Unfortunately, most people who view this chart and see the numbers and think, "Oh my God. Even Tokyo is under heavy radioactive fallout." The way this chart is drawn makes it look like all of Japan is under the area of dangerous radioactivity.

Look at Tokyo. It says, "0.064" microsieverts per hour. That sounds terrible, doesn't it? But what does that mean? Most people won't bother to think about it and will fear.

Let me give you an example of what this means.

Tokyo is experiencing 0.064 microsieverts per hour radiation. The typical one-way flight from Narita airport to New York is 190 microsieverts. If we must fully understand and gauge and compare our risk in Tokyo, then we must understand these terms and their relation.

So let's calculate the risk. The typical one-way flight from Narita airport to New York is 190 microsieverts. 

Now Tokyo is experiencing 0.064 microsieverts per hour radiation. Let's equalize that with our flight. What is our risk?

To calculate this very simple problem we divide 190 by 0.064. This equals 2,968.75 microsieverts. Divide that by 24 hours in a day equals 123 days. That means that the dose of radiation you get from living in Tokyo for 123 days is equal to the dose of radiation you get from one one-way flight from Tokyo to New York. 

- Thanks to Marilyn Vos Savant for the Monty Hall Dilemma in "The Power of Logical Thinking"


NOTE: I want to be the first one to admit that I had something wrong, or make a mistake. When I first posted this, I had miscalculated the radiation on the flight to New York. The flight to New York is 190 microsieverts per hour. It takes 10 ~ 11 hours to fly to New York. I had it calculated at 190 microsieverts total. So the amounts above are recalculated to show this error on my part. Sorry for the trouble.