[Roger Wattenhofer told me this puzzle.]
Someone picks, at their will, two cards from a deck of cards. The cards have different numbers, so one is higher than the other. (In other words, the person picks two distinct numbers in the inclusive range 1 through 13.) The cards are placed face down on a table in front of you. You get to choose one of the cards and turn it face up. Now, you will select one of the two cards (one of whose face you can see, the other one you can’t). If you select the highest card, you win. Design a card-selection strategy for which your chance of winning is strictly greater than 50%.