Draw three rectangles on a piece of paper. Those represent doors. Above those rectangles put C, G, and G.
Put a giant X through one of the doors and one of the G’s. You are subtracting (1) from the supposedly possible (2) G’s and (1) from supposedly (3) options of doors.
Monty crosses off a door and a goat. You have to subtract 1 from the class of object “door” and 1 from the class of the type “goat”.
Goat and Door are two different variables. They exist independently but are combined in the same way we use containers in programming.
Example:
Door C = Goat (2)
You have two objects you’re dealing with.
Your goal is to get a car. There are only one of those.
You don’t have to worry about a second goat or a third door being an option. The problem lies in information transference because both groups are combining objects.
Simply put:
Stage 1:
3 Doors
2 object types (car or goat)
2 possible objects (car or 1 of 2 goats)
Stage 2:
2 Doors
2 object types (car or goat)
2 actual (now defined objects (car and 1 goat)
At stage 1 either goat is possible to end up with, as well as the car. By stage 2, you will have defined that 1 of the goats is not possible to pick. Monty removes it from the pool of options. In addition, he removes the door as well.
In fractions you don’t just change the enumerator, you change the denominator. The denominator represents possible choices and the enumerator represents quantity of choices or guesses.
Simply put: Stage 1 and Stage 2 are not the same. He is not there to reveal information. We already knew one of the remaining doors contained a goat. The location doesn’t help you solve anything whether its on the right or left side. He reveals nothing.
Why is he there?
Only to eliminate you having to choose from a pool of 3 doors, because prior to this each door had unknown contents.
2 total guesses (“stay” and “switch”), but only one of those guesses will be revealed to your eyes.
Monty is saying:
2 guesses (1 unrevealed) out of 3 doors you can pick…no, let me make this easier. Ill take a door and goat both and you choose from the remaining pool.
The problem is actually simple if you understand why we use fractions to begin with.
Amount of guesses/Possible prizes
Use that formula in each stage (as information is revealed). Then you will see that Stage 1 and Stage 2 are entirely separate fractions.
When I said both groups have a problem:
In Stage 1, you have to assume your first choice is a goat (2/3 chance). You should switch immediately according to probability logic.
In Stage 2, you no longer have to worry about those same odds because the goat & door which were previously half the quantity of your entire loss probability (1/2 x 66.6 percent) has nnow been eliminated from a possible decision.
That number is evenly distributed to the 2 remaining mystery doors. (16.67 percent to each)
You had 3 decisions originally = 3 doors = bottom denominator.
You had 1 guess originally. Monty never substantiates the results of this guess. He only gives you the option of making a new guess, which is the same as this guess being the first one. You never satisfied the results of this original guess, so it was only a placeholder to get you to Stage 2. It wasn’t even a real guess because nothing is final until it is compared against the opposing information.
Trying to link the two guesses together is where people are having a problem. The paradox is that Monty is changing the game into a “new” game each time.
To better illustrate, imagine you are on “Who wants to be a Millionaire” and you have (4) possible answers. It doesn’t matter how many times you change your guess until Regis says:
“Is that your final answer?”
None of those original guesses will ever be submitted against the correct answer. Only your final selection has any bearing on whether you win.
In Stage 1 of Monty’s game (we should actually call it “Game 1”) our supposed “guess” doesn’t matter. It’s not a real guess, it only exists if and when we follow through with it. We can change our mind 1,000 times but only the “opening of the door” seals that conceptual guess into an actual guess…in the same way that Regis revealing an answer seals our guess on his show.
The revelation is the point of no return. Assuming that the “two” Stages are “one” game and are interrelated is only complicating solving the problem.
The math is this problem is very basic math which we all learned in elementary school. However, the conceptual science is advanced and thus we are manipulating the wrong digits to achieve our answer.
Put another way: 10 coin tosses are 10 quantities of (1) coin toss each. There is no need to group them together. Humans created this grouping by concepts such as “best out of 10”. Each is independent and the only thing achieved by comparing to another flip is “averages”.
Averages in themselves only exist once we define the quantity of however many things we are going to compare. All of these concepts are interrelated.
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