Game Show Problem

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Re: Game Show Problem

Postby Gofer » Thu Aug 31, 2017 4:37 pm

Robert, you are wasting your time trying give reasons for why we can deduce (as in "logically follows") the correct answer from the problem description when there isn't one, the reason being it isn't clear whether such a description refers to a happenstance or event or to actual rules of the game, and even if it were, such as the host always opening a goat-door, we couldn't rule out it applying to all goats, even those initially selected by the contestant, which would then instead imply the answer 1/2 to the problem.
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Re: Game Show Problem

Postby robert 46 » Fri Sep 01, 2017 7:43 am

Gofer wrote:
> Robert, you are wasting your time trying give reasons for why we can deduce (as in
> "logically follows") the correct answer from the problem description when there
> isn't one, the reason being it isn't clear whether such a description refers to
> a happenstance or event or to actual rules of the game,

Even if it is a one-time-occurrence, never to be repeated, we can work out a probability. We just assume the opportunity to switch was spontaneously given without bias.

> and even if it were [the rules of the game], such
> as the host always opening a goat-door, we couldn't rule out it applying to all
> goats, even those initially selected by the contestant, which would then instead
> imply the answer 1/2 to the problem.

We rule out the host revealing the player's door has a goat because it either does not offer the opportunity to switch, or winning the car is a certainty- not requiring a probability calculation [1].

We are tasked with finding an answer which is superior to all others. We use information provided and reasonable assumptions implied by the problem statement. For example:
The problem does not say the host deliberately reveals a goat, but the fact that the host knows where the prizes are implies that the host knows what is behind the door which will be opened before it is opened; so this lack of uncertainty as to what will be revealed can be interpreted as implying deliberate action.


[1] Consequent to the reduction of getting the higher value prize behind the remaining doors.
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Re: Game Show Problem

Postby Gofer » Wed Sep 06, 2017 8:38 am

Robert, I'm condensing your theory to a single statement:

Pr(HS|Pc) = x = Pr(HS|Pg), for some 0 < x < 1,

where HS is the host opening a goat-door other than the player's and offer a switch, and Pc is the player initially choosing the car-door, and Pg is the player initially choosing a goat-door,

which basically says that the host action, as far as revealing a goat and offer a switch, is independent of the player action.

But nothing in the problem description allows us to make such a deduction, which can only be made by appealing to reasonable assumptions.

The host knowing the location of the car does nothing for determining the distribution of the random variable associated with him.
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