Problem Statement 
 Taro likes apples very much. He has N boxes numbered from 0 to N1. There are K different types of apples numbered from 0 to K1. You are given three String[]s, hundred, ten and one. Concatenate jth characters in hundred[i], ten[i] and one[i] in this order to get a string that represents the number of jth type apples in Box i (it may have leading zero(s)). This number will be between 0 and 199, inclusive.
He decided to choose one apple from his boxes, and he does so in the following way:

First Step: He chooses a nonempty subset of his N boxes randomly and transfers all apples from those boxes to another box (this is a box other than the original N boxes and it is initially empty). Each nonempty subset of boxes has the same probability of being chosen.

Second Step: He chooses one apple from the new box randomly. Each apple in the box has the same probability of being chosen.
Return a double[] that contains exactly K elements and whose ith element is the probability that Taro chooses an ith type apple.


Definition 
 Class:  RandomApple  Method:  theProbability  Parameters:  String[], String[], String[]  Returns:  double[]  Method signature:  double[] theProbability(String[] hundred, String[] ten, String[] one)  (be sure your method is public) 




Notes 
  Your return value must have an absolute or relative error less than 1e9. 

Constraints 
  N will be between 1 and 50, where N is the number of elements in hundred. 
  K will be between 1 and 50, where K is the number of characters in hundred[0]. 
  ten and one will contain exactly N elements. 
  Each element in hundred, ten and one will contain exactly K characters. 
  Each character in hundred will be '0' or '1'. 
  Each character in ten and one will be a digit ('0''9'). 
  Each box will contain at least one apple. 

Examples 
0)  
  Returns: {0.38461538461538464, 0.6153846153846154 }  There is only one box which contains 5 type0 apples and 8 type1 apples. The probability of choosing a type0 apple is 5 / 13.



1)  
 {"00", "00"}  {"00", "00"}  {"21", "11"} 
 Returns: {0.5888888888888889, 0.4111111111111111 }  If he chooses only box 0 in the first step, the probability of choosing a type0 apple is 2 / 3.
If he chooses only box 1 in the first step, the probability of choosing a type0 apple is 1 / 2.
If he chooses both boxes in the first step, the probability of choosing a type0 apple is 3 / 5.
So the probability of choosing a type0 apple is (2 / 3 + 1 / 2 + 3 / 5) / 3 = 53 / 90.



2)  
 {"0000", "0000", "0000"}  {"2284", "0966", "9334"}  {"1090", "3942", "4336"} 
 Returns:
{0.19685958571981937, 0.24397246802233483, 0.31496640865458775, 0.24420153760325805 }  

3)  
 {"01010110", "00011000", "00001000", "10001010", "10111110"}  {"22218214", "32244284", "68402430", "18140323", "29043145"}  {"87688689", "36101317", "69474068", "29337374", "87255881"} 
 Returns:
{0.11930766223754977, 0.14033271060661345, 0.0652282589028571, 0.14448118133046356, 0.1981894622733832, 0.10743462836879789, 0.16411823601857622, 0.06090786026175882 }  

4)  
  Returns: {1.0, 0.0 }  One box with 100 type0 apples and no type1 apples. 

