Problem of the Week 950

Here's the Deal

Alice: Here's the deal. You give me $10. Then I will deal four cards (from a regular 52 card deck), chosen randomly, face down. You get to look at #1 first and decide whether to keep it. If not, look at #2 and decide whether to keep that one. If not look at #3, and decide. If you don't take that, then #4 is your choice. If your chosen value is n, I will pay you $n. Then we can reshuffle the entire deck, you give me another $10, and we can play again, and again, and again.

Bob: Hmmm....I need a good strategy to beat you at this game, but I think I can do it.

Help Bob out with a strategy that will win. Note that the cards all have face value except face cards, which have the values Jack = 11, Queen = 12, and King = 13.

Source: Stan Wagon.