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- Feb 28, 2001

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Thank you, @Cronk.

@Jon - here is the updated game results grid, this time exhaustive and with further analysis.

I repeat the game rules just so we agree (and if you believe I have misstated a rule, PLEASE be specific.)

1. Car placement: The car IS behind one of three doors. A non-prize (zonk) is behind each of the other two doors. In the updated grid, the car placement is in the "Car" column.

2. Pick: You select one of the three doors. Your selection is in the "Pick" column.

3. Reveal: Monty reveals to you the contents of a door you did not pick, and this door WILL be a zonk. The door chosen for the reveal is listed in the "Reveal" column.

However, the "reveal" choice is constrained by these rules:

3.a - Monty will never reveal the door you picked

3.b - Monty will never reveal the door hiding the car (see "Resolution" later.)

3.c - IF and ONLY IF you picked the correct door, Monty can choose which of two zonks he will show you and those choices have equal probability. Since you have no way of knowing WHY Monty revealed a door, you cannot draw an inference for why one was picked vs. another. I.e. it is questionable to apply the principle of restricted choice since you don't know whether Monty HAD a choice. (In fact, more often than not, he DOES NOT have a choice.)

3.d - Because of rules 3.a and 3.b, certain combinations of picks cannot occur - and there is a 0 in the "Possible" column for those cases, along with a comment in the "comments" column as to which rule was violated. NO statistics can be derived from rows that violate game rules. I have also highlighted the "forbidden" cases in light yellow.

4. Switch: You now get the offer to switch from the "Pick" door to the remaining door that was neither picked nor revealed. In the PDF/Spreadsheet, the "Switch" column is 1 if you switched or 0 if you stayed with your original choice.

5. Resolution: You either WIN or LOSE the car. If you won, it was either because you stayed or switched. The four remaining columns of the PDF/Spreadsheet show you wins and losses and also show you wins because of switching and wins because of staying.

Analysis:

You have 3 doors for a car, 3 doors for a pick, 3 doors for a reveal, and 2 choices (stay/switch). That is 3 x 3 x 3 x 2 = 54 combinations.

There are 54 rows in the spreadsheet, one row for each possible combination.

You have 27 cases where you switched (= 54/2) and 27 where you stayed (0 in the Switch column)

The car appears behind each door 18 times (=54/3).

Each possible pick appears 18 times.

Each possible reveal (regardless of the rules) appears 18 times.

All combinations of Car, Pick, Reveal, and Switch have been represented, so this is an exhaustive layout.

The rules block cases where the reveal overlaps either the pick or the car location. A reveal can be any one of 3 numbers, so you would expect this overlap to occur 54/3 or 18 times. In the grid there are 18 cases where the reveal overlaps the car location. There are also 18 cases where the reveal would overlap the pick location. Of these, six cases break both rules at once. 18 + 18 = 36. then 36 - 6 overlaps = 30 impossible cases, leaving 24 cases out of the 54 possible combinations. The sums of "Possible" rows show that you have 24 possible situations. The rules have eliminated all other cases. This result therefore is consistent.

The sums for wins and losses are 12 and 12 so all 24 situations are accounted for, showing you have equal odds of winning or losing. The sums for wins by staying vs. wins by switching are 6 and 6 so all 12 winning situations are accounted for, and you have equal odds for staying or switching.

@Jon - here is the updated game results grid, this time exhaustive and with further analysis.

I repeat the game rules just so we agree (and if you believe I have misstated a rule, PLEASE be specific.)

1. Car placement: The car IS behind one of three doors. A non-prize (zonk) is behind each of the other two doors. In the updated grid, the car placement is in the "Car" column.

2. Pick: You select one of the three doors. Your selection is in the "Pick" column.

3. Reveal: Monty reveals to you the contents of a door you did not pick, and this door WILL be a zonk. The door chosen for the reveal is listed in the "Reveal" column.

However, the "reveal" choice is constrained by these rules:

3.a - Monty will never reveal the door you picked

3.b - Monty will never reveal the door hiding the car (see "Resolution" later.)

3.c - IF and ONLY IF you picked the correct door, Monty can choose which of two zonks he will show you and those choices have equal probability. Since you have no way of knowing WHY Monty revealed a door, you cannot draw an inference for why one was picked vs. another. I.e. it is questionable to apply the principle of restricted choice since you don't know whether Monty HAD a choice. (In fact, more often than not, he DOES NOT have a choice.)

3.d - Because of rules 3.a and 3.b, certain combinations of picks cannot occur - and there is a 0 in the "Possible" column for those cases, along with a comment in the "comments" column as to which rule was violated. NO statistics can be derived from rows that violate game rules. I have also highlighted the "forbidden" cases in light yellow.

4. Switch: You now get the offer to switch from the "Pick" door to the remaining door that was neither picked nor revealed. In the PDF/Spreadsheet, the "Switch" column is 1 if you switched or 0 if you stayed with your original choice.

5. Resolution: You either WIN or LOSE the car. If you won, it was either because you stayed or switched. The four remaining columns of the PDF/Spreadsheet show you wins and losses and also show you wins because of switching and wins because of staying.

Analysis:

You have 3 doors for a car, 3 doors for a pick, 3 doors for a reveal, and 2 choices (stay/switch). That is 3 x 3 x 3 x 2 = 54 combinations.

There are 54 rows in the spreadsheet, one row for each possible combination.

You have 27 cases where you switched (= 54/2) and 27 where you stayed (0 in the Switch column)

The car appears behind each door 18 times (=54/3).

Each possible pick appears 18 times.

Each possible reveal (regardless of the rules) appears 18 times.

All combinations of Car, Pick, Reveal, and Switch have been represented, so this is an exhaustive layout.

The rules block cases where the reveal overlaps either the pick or the car location. A reveal can be any one of 3 numbers, so you would expect this overlap to occur 54/3 or 18 times. In the grid there are 18 cases where the reveal overlaps the car location. There are also 18 cases where the reveal would overlap the pick location. Of these, six cases break both rules at once. 18 + 18 = 36. then 36 - 6 overlaps = 30 impossible cases, leaving 24 cases out of the 54 possible combinations. The sums of "Possible" rows show that you have 24 possible situations. The rules have eliminated all other cases. This result therefore is consistent.

The sums for wins and losses are 12 and 12 so all 24 situations are accounted for, showing you have equal odds of winning or losing. The sums for wins by staying vs. wins by switching are 6 and 6 so all 12 winning situations are accounted for, and you have equal odds for staying or switching.

*Quod erat demonstradum.*