Suppose there are three doors
Door 1 Door 2 and Door 3
Suppose our strategy is when the referee opens a door for the second time we will not switch our first choice
Let us assume that million dollar prize is behind the door 2
Now, you will have three cases
You choose door 1 and the referee opens door 3
Since you are not switching you will lose.
You choose door 2 and the referee opens door 1 or door 3
In this case you win.
You choose door 3 and referee opens door 1
In this case you lose
So, your chance of winning is (1/3) with the strategy of not switching.
Now, suppose your strategy is that you will switch the door when the refree opens the door for the second time.
This time also you will have three cases
Case 1 You choose first door referee opens third door
Since you are switching you win.
Case 2 You choose second door referee opens door 1 or door 3
In this case you will lose
Case 3 You choose third door referee opens door 1
In this you will win since you are switching
So, the chance of winning when you are switching is 2/3.
So, if I have another choice I will definitely switch.