While I admit I am no supreme expert in statistical analysis, I have to say that I'm seeing a trend of incorrect thinking here.

The way this should be defined is as follows:

Decision tree 1:

C = Car
G1 = Goat 1
G2 = Goat 2

Possibilities for Decision Tree 1:

C G1 G2
G1 C G2
G1 G2 C

The chances of picking the correct door at random are 1 in 3.

The contestant picks door number 1. They have a 1 in 3 chance of having picked the correct door.

Monty opens door X and shows the contestant a goat. Monty then offers to let the contestant choose another door.

This creates a NEW decision tree, where the possibilities are:

C G1
G1 C

This is an original decision and is not linked in any way other than chronology and prize to be won to the previous decision.

Thusly, there is a 50% chance to select the correct door.

In this case, staying and changing are equally possible, and are equally beneficial.

Note that when considered as a separate decision, the odds of matching are different.

Compare this to matching dice. The odds of rolling two six sided dice simultaneously and coming up with the same number are far different than the odds of rolling a single six sided die, recording the first number, and then rolling a second six sided die hoping to match that first number. In this case, the second option is far more likely (1 in 6 rather than 1 in 36).

Subbie