Program logic bugs vs input/environmental errors

[Nick Sabalausky via Digitalmars-d]

On 10/07/2014 08:37 PM, H. S. Teoh via Digitalmars-d wrote:
> On 10/08/2014 12:10 AM, Nick Sabalausky wrote:
> [...]
>> I've managed to grok it, but yet even I (try as I may) just cannot
>> truly grok the monty hall problem. I *can* reliably come up with the
>> correct answer, but *never* through an actual mental model of the
>> problem, *only* by very, very carefully thinking through each step of
>> the problem. And that never changes no matter how many times I think
>> it through.
> [...]
>
> The secret behind the monty hall scenario, is that the host is actually
> leaking extra information to you about where the car might be.
> [...]

Hmm, yea, that is a good thing to realize about it.

I think a big part of what trips me up is applying a broken, contorted 
version of the "coin toss" reasoning (coin toss == my usual approach to 
probability).

Because of the "coin toss" problem, I'm naturally inclined to see past 
events as irrelevant. So, initial impression is: I see two closed doors, 
an irrelevant open door, and a choice: "Closed door A or Closed door B?".

Obviously, it's a total fallacy to assume "well, if there's two choices 
then they must be weighted equally." But, naturally, I figure that all 
three doors initially have equal changes, and so I'm already thinking 
"unweighted, even distribution", and then bam, my mind (falsely) sums 
up: "Two options, uniform distribution, third door isn't a choice so 
it's just a distraction. Therefore, 50/50".

Now yes, I *do* see several fallacies/oversights/mistakes in that, but 
that's how my mind tries to setup the problem. So then I wind up working 
backwards from that or forcing myself to abandon it by starting from the 
beginning and carefully working it through.

Another way to look at it, very similar to yours actually, and I think 
more or less the way the kid presented it in "21" (but in a typical 
hollywood "We hate exposition with a passion, so just rush through it as 
hastily as possible, we're only including it because writers expect us 
to, so who cares if nobody can follow it" style):

1. Three equal choices: 1/3 I'm right, 2/3 I'm wrong.

2. New choice: Bet that I was right (1/3), bet that I was wrong (2/3)

3. "An extra 33% chance for free? Sure, I'll take it."

Hmm, looking at it now, I guess the second choice is simply *inverting* 
you first choice. Ahh, now I get what the kid (and you) was saying much 
better: Choosing "I'll change my guess" is equivalent to choosing *both* 
of the other two doors.

The fact that he opens one of those other two doors is a complete 
distraction and totally irrelevent. Makes you think you're only choosing 
"the other ONE door" when you're really choosing "the other TWO doors". 
Interesting.

See, this is why I love this NG :)