Comment by nobaelazum

Some explanations of this problem are far too complicated. This is how I explain it.

1/3 of the time, you guess correctly on your initial guess. If you switch, you'd be wrong.

2/3 of the time, you guess incorrectly on your initial guess. If you switch, you're right.

So when switching, the expecting outcome 1/3 of the time is 0, but the expected outcome 2/3 of the time 1. (1/3)0 + (2/3)1 = 2/3