Monday, May 16, 2011

Why the Three Doors is Not Such a Difficult Problem

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


James said...

Oooh, oooo, oooooo! Mr Kotter, Mr Kotter!

Lessee, first take the chicken across. Come back, pick up the seeds and bring them back across. When you drop off the seeds, put the chicken back in the boat before he eats them. Take the chicken back across and then drop off the chicken and pick up the dog. Bring the dog across and leave it with the seeds, then go back across and pick up the chicken.

mikeintokyorogers said...

James! An A+ and extra points for enthusiasm!

Poots said...

Not so fast Mike.

You are playing a bit fast and loose with the options here.

In your chart options 2 and 5 are NOT offered.

You do not allow for the option of choosing what is a 100% chance of opting for the correct door, you only allow for it to be eliminated from the selection.

Since options 2 and 5 are eliminated that only leaves options 1, 3, 4, & 6.

In each case – switch from door 1 or stay with door 1 the chances are the same – 50/50.

mikeintokyorogers said...

Poots. Thanks. Read it again. In the first three you choose door #1 and switch. The host always shows you a door with the sandwich.
In the second examples doors 4~6. You stay. It's 2/3 to switch.

mikeintokyorogers said...

"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."

Poots said...

OK, I went back and re-read it.

It says that I am shown door B or 2, NOT that I am always shown the sandwich.

That makes a big difference. Always door 2...

boo said...

Oh my...please don't compare mathematicians to economists and politicians...please. It upsets my stomach.

In fact, in your previous post you made a very good point: that a certain condition was implied but not stated outright because you thought it was "obvious" as a TV guy.
This points to a fundamental difference between natural language, which relies on unstated external context for meaning, versus mathematical language, which does its best to eliminate external context.
This is why I really really hate it when people state math problems in English. It confuses the matter, and clever pols/economists take advantage of unstated "obvious" assumptions to get people to think the way they want, while sounding sort of scientific. The use of English is the equivalent of the distractor in the shell game.

mikeintokyorogers said...

OK, Boo... So what you are saying is that in 9 times out of ten that you refrain from voting that had you voted you would have chosen the logical choice.... Me!
Thanks I knew I could count on you in the next election! Chuckle! PS: Looking for a running mate, interested?