How is the probability of winning a block calculated from the difficulty?
If the current difficulty is D, then the target hash (the value below which block hashes must go) is:


  0x00000000FFFF0000000000000000000000000000000000000000000000000000 / D


(by definition of difficulty, which is a fraction of the maximum target), or otherwise put, the number of valid hashes is:


  65535 * 2208 / D


Which means that the ratio of all hashes over valid hashes would be:


  2256 / (65535 * 2208 / D) =
  D * 248 / 65535 =
  D * 4295032833


Which, at the current (January 2017) difficulty of D =  392963262344.3704 means that one hash in


  392963262344.3704 * 4295032833 = 1687790113931869416948


results in a valid block, or each attempt has a chance of


  1 / 1687790113931869416948 = 0.000000000000000000059 %