What is the branching factor for nondeterministic games?
The branching factor depends on exactly how you model the game.  A standard way to model games that have some randomness is to imagine adding another player (the "dealer") whose only purpose is to make a random choice. 

So if your game involves only one player Alice, we'll add another player Declan (the dealer).  Alice and Declan alternate turns.  During her turn, Alice can either play one of her cards or choose to draw from the pile.  Thus, if Alice has $n$ cards, she has $n+1$ actions she can choose from.  During Declan's turn, if Alice previously chose to draw, Declan selects one of the remaining cards and adds it to Alice's hand; otherwise he does nothing.  Thus, if there are $m$ cards remaining in the pile, Declan has either $m$ actions he can choose from, or only one.

With this model, the maximum branching factor is 53 (assuming Alice can have up to 52 cards in her hand).