At a high level this is confusing because of the idea of informative and conditioning. As humans we're naturally used to explaining things based upon events that have happened this is a hack that enables us to process information. Unfortunately it backfired in cases such as these.

The probability of it being behind your door (1/3) or the other doors (2/3) doesn't change, it's independent of the events. When a door is opened we know that it is not behind one door so the 2/3 probability now only has a degree of freedom of 0 whereas before it has a df of 1. So the offs haven't changed but the df has.

Here's another way to look at it. The average probability does change however (1/3 + 2/3)/2 --> 1/2. This differs from what it was at the beginning (1/3+1/3+1/3)/3-->1/3 because there are fewer degrees of freedom.

So notice there at first there were N-1 or 2 degrees of freedom for the entire context. And N-2=1 for the other doors. Opening a door removes the only degree of freedom for the subgroup. It is this added constraint that increases the rate at which you can infer about this subgroup, especially upon the last guess. It has no more degrees of freedom to vary.

At a high level this is confusing because of the idea of informative and conditioning. As humans we're naturally used to explaining things based upon events that have happened this is a hack that enables us to process information. Unfortunately it backfired in cases such as these.

The probability of it being behind your door (1/3) or the other doors (2/3) doesn't change, it's independent of the events. When a door is opened we know that it is not behind one door so the 2/3 probability now only has a degree of freedom of 1 whereas before it has a df of 2. So the offs haven't changed but the df has.

Here's another way to look at it. The average probability does change however (1/3 + 2/3)/2 --> 1/2. This differs from what it was at the beginning (1/3+1/3+1/3)/3-->1/3

So notice there at first there were 3 degrees of freedom (2 in the other doors) now there are only 2 (1 less behind the other doors)