It's a 1/3 chance if you pick before the second door is opened and do not switch.
It's a 1/2 chance if you pick after the second door is opened, regardless of switching.
It's a 2/3 chance if you pick before the second door is open and switch after the second door is opened.
I recommend reading again, because I don't want to write up the logic.
Of a bigger concern is that this was not true on "Let's Make a Deal". The problem assumes that Monty always offered the option to switch. In reality, he did not, and the offers were not random. That corrected for the issue to a large degree in practice.
It's a 1/2 chance if you pick after the second door is opened, regardless of switching.
It's a 2/3 chance if you pick before the second door is open and switch after the second door is opened.
I recommend reading again, because I don't want to write up the logic.
Of a bigger concern is that this was not true on "Let's Make a Deal". The problem assumes that Monty always offered the option to switch. In reality, he did not, and the offers were not random. That corrected for the issue to a large degree in practice.