Program logic bugs vs input/environmental errors

[H. S. Teoh via Digitalmars-d]

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.

You make a first choice, which has 1/3 chance of being right, then the
host opens another door, which is *always* wrong. This last part is
where the information leak comes from.  The host's choice is *not* fully
random, because if your initial choice was the wrong door, then he is
*forced* to pick the other wrong door (because he never opens the right
door, for obvious reasons), thereby indirectly revealing which is the
right door.  So we have:

1/3 chance: you picked the right door. Then the host can randomly choose
	between the 2 remaining doors. In this case, no extra info is
	revealed.

2/3 chance: you picked the wrong door, and the host has no choice but to
	pick the other wrong door, thereby indirectly revealing the
	right door.

So if you stick with your initial choice, you have 1/3 chance of
winning, but if you switch, you have 2/3 chance of winning, because if
your initial choice was wrong, which is 2/3 of the time, the host is
effectively leaking out the right answer to you.

The supposedly counterintuitive part comes from wrongly assuming that
the host has full freedom to pick which door to open, which he does not
in the given scenario. Of course, this scenario is also often told in a
deliberately misleading way -- the fact that the host *never* opens the
right door is often left as an unstated "common sense" assumption,
thereby increasing the likelihood that people will overlook this minor
but important detail.


T

-- 
Written on the window of a clothing store: No shirt, no shoes, no service.