The following games are played with 8 cards: the four kings and the four aces. The dealer fans the cards with their back to you and asks you to take one, but not to look at it. He tells you that you will win a prize if your chosen card is an ace. There are two versions of what happens next.
(a) In version 1, he closes up the remaining 7 cards, shuffles them and deals the top card face up on the table, and it is seen to be the king of spades. He then deals the next card face down on the table. He then offers you the chance to trade the card you hold for the face-down card that was just dealt. So you have two choices: stay or switch. What are your chances of winning under each choice?
(b) In version 2, he announces that he is going to select a king from the remaining 7 cards, and he selects the king of spades and lays it face up on the table. He then shuffles the remaining six cards and deals the top card face down on the table, and offers you the chance to trade the card you hold for this new card. So you again have two choices: stay or switch. What are your chances of winning under each choice?