# Sample space of the Monty Hall problem

I have read two sources where the outcomes of the Monty Hall problem are triples of the form:

(initial guess, he curtain Monty opens, the curtain where the car is)

However, I think it should be:

(initial guess, the curtain Monty opens, switched curtain, the curtain where the car is)

Is the sample space consisting of these outcomes wrong? It seems that it is because I am getting that it doesn't matter if one switches curtains.

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If the contestant chooses at random,$2/3$ of the time s/he will choose a goat. Then Monty will show a goat in another door, meaning changing doors will give you the car. Only in $1/3$ of the time, will a contestant select the door containing the car; in this last case, the contestant loses by changing the choice of doors. This means $2/3$ of the time, you're better-off by changing doors.