bet a dollar to win a dollar
3 cups are turned upside down, one has a ball under it
if you pick the ball you win
I know which cup has the ball under it
you have a 1/3 chance of picking the cup with the ball
you pick a cup
I turn over one of the other cups to show you that it is empty and remove that cup
I give you the choice of keeping your cup or switching to the remaining cup
It's obvious at this point I am trying to give myself a chance to beat you. If you picked an empty cup I would have just asked you to turn over your cup and the game is over with me winning.
in a game like that your chances of winning are 1/3 at the start, if there is a second round your chances are 100% if you keep your cup and zero if you switch
things get complicated if we change the rules to where there had to be a second round.
1/3 at the start and then 50/50 for round two, giving you an overall chance of 1/6
with the 'let's make a deal' example the host controls the rules, knows where the prize is and remember, his goal is not to make a profit but to make the game entertaining for the viewers.
if the host has 2 or 3 rule sets to choose from during each round, and can add or subtract rounds then it becomes a game of calling someones bluff