Good start, but your truth table is incomplete. You have removed the non-sensical rows in which Monty opens the door showing the car and you either switch or don't switch. It would seem that this doesn't matter, but it is skewing the results of your truth table because every line of a truth table is equally weighted. A truth table represents all possible inputs and then displays all logical (but not neccessarily "sensical") outputs based on some application of an operation.
Here is an example. Assume we have a random number generator, 0 though 7. What is the probability of any of the outcomes? 1/8. There are 8 rows to the truth table, which results all possible numbers generated in our system.
000
001
010
011
100
101
110
111
What is the probability of an even number? 4/8 = 1/2. What is the probability of an odd number? 4/8 = 1/2. So far, so good.
Now, let's apply a rule that says that any time the system generates an even number (include 0), we add 1 to make it odd (formally, the operation is outcome=input|0x001). The probability of the outcome of any given even number is now 0/8 and an odd number is now 2/8 = 1/4. The truth table still has 8 lines, but because of the rule applied to the initial uniform distribution of inputs, we no longer have a uniform distribution for the outcomes.
Note that I show the non-trivial application of the operation as ->
000 -> 001 = Odd
001 = Odd
010 -> 011 = Odd
011 = Odd
100 -> 101 = Odd
101 = Odd
110 -> 111 = Odd
111 = Odd
The initial uniform distribution has been altered (weighted) in favor of a new distribution by modifying certain random input to become an different outcome.
If one were to reduce the truth table by not showing the altered line, the truth table would be:
001 = Odd
011 = Odd
101 = Odd
111 = Odd
which clearly does not show all the initially generated values, but still gives an accurate probability due to the uniform application of an essentially linear altering / weighting operation.
Now, to match this up with the Monty Hall Problem more closely, change the modification rule to only add 1 to even numbers greater than or equal to 4 (if(input>=0x04) then outcome=input|0x01). This operation is applied uniformly, but the operation itself has a non-linearity in it (only greater than or equal to 4). Here is the new truth table showing the non-uniform distribution of outcomes based on a uniform distribution of inputs when a non-linear operator is applied:
000 = Even
001 = Odd
010 = Even
011 = Odd
100 -> 101 = Odd
101 = Odd
110 -> 111 = Odd
111 = Odd
Now, the probability of getting an even number is 2/8 = 1/4 and the probability of getting an odd number is 6/8 = 3/4, despite the initial uniform distribution. If the truth table is reduced simply to outcomes by collapsing input rows affected by the non-linear operation into input rows with the same output, then a deceptive truth table, similar to the one in the above post, is created:
000 = Even
001 = Odd
010 = Even
011 = Odd
101 = Odd
111 = Odd
In this case, it appears that the probability of there being an odd number is simply 4/6 = 2/3 and even is 2/6 = 1/3, which is incorrect, since our number generator was uniform, and the rule altered the 2 out of 8 of the initial numbers to become something else. The new truth table is no longer uniform in distribution because it is no longer representing all input rows.
In the truth table in the above post, only 8 outcomes are shown from an actual truth table of 12 outcomes (using only CGG, but similar for other permutations). 4 input rows have been eliminated, thus the error.
The actual truth table is (N=NO, Y=YES for the switch column):
CGG12N = Win
CGG12Y = Lose
CGG12N = Win
CGG12Y = Lose
CGG21N -> CGG23N = Lose (this row was collapsed above)
CGG21Y -> CGG23Y = Win (this row was collapsed above)
CGG23N = Lose
CGG23Y = Win
CGG31N -> CGG23N = Lose (this row was collapsed above)
CGG31Y -> CGG23Y = Win (this row was collapsed above)
CGG32N = Lose
CGG32Y = Win
With a uniform truth table, showing the application of the weighting, the correct probabilities are found. There are 6 ways to win and 6 ways to lose, but only 2 of 6 ways to win do not switch (CGG12N and CGG13N). The other 4 ways to win in the truth table required switching (CGG21Y->CGG23Y, CGG23Y, CGG31Y->CGG32Y, CGG32Y).
Because Monty's operation on the doors is non-linear (random choice of two goat doors when user has selected the car and a forced single selection of a goat door when the user has selected a goat), certain outcomes in a reduced (collapse altered input rows) truth table will have non-linear weighting when determining probability.
The point is that any time you use a truth table you must show all inputs (inclusive of non-sense) and all outcomes if you are trying to count up the outcomes and determine the probability of a particular outcome.
Alternatively, which I will just touch on, you cannot collapse an operation into the input columns, which includes the reveal and switch operations, but not in other cases. From a system perspective, the shuffling of C, G, and G, and the initial selection of a door is one system of inputs, with an outcome (win / lose) based on an equivalance operation. Based on the output of that system and the state of the inital system, Monty then applies one of two different operation ("pick a door of 2 goat choices randomly" or "pick the remaining goat") to create two different sets of outcomes. By lumping them together without accounting for the non-linearity of Monty's two different operations, the end-to-end truth table is incorrect.
