this is the best answer

> The person who changes his choice will win if and only if his first choice was wrong, and there is a probability of 2/3
on that.
>
> The person who does not change his choice will win if and only if his first choice was right. There is a probability of 1/3
on that.

both decisions are based on information you don't have, and the average of 1/3 and 2/3 is 1:2

it's a coin flip either way, unless you know you were right in which case you hold, if you know you were wrong you switch

you don't know this, so this doesn't change the odds since it is 100% certain that one is right and one is wrong

the average of the odds described in this answer average to 50/50, literally a coin flip

flipping a coin to decide won't change that, as this information is not known to the host either

the only thing that could be at play is the host is trying to read you and tries to use all kinds of unconscious signals to provoke you to change when they know you were right