Thursday, 29 June 2017

To Maximize the Probability of Red Marble

You have 50 red balls and 50 blue balls, you have to place these balls in two containers in such a way that probability of a person picking up a red ball from any container is maximum.

1. For container 1,
You should have to place one red marble into 1st Container. By this when we select this Container 1 P(Red|C1) will be 1.

2.  For container 2,
Rest all marbles should have to place into 2nd container. By this when we select this Container 2 P(Red|C2) will be 49/99.

Here, P(C1) = P(C2) = 1/2 as there is only 2 containers.

So answer will be P(Red) = P(C1)*P(Red|C1) + P(C2)*(Red|C2)
                                                  P(Red) = (1/2)*(1) + (1/2)*(49/99)
                                                  P(Red) = 0.747474.

Proof of above given solution using the Java code:
public class MaximizeRedBallProbability {
     /**
      * This is the code which maximize the probability.
      * @param red
      * @param blue
      * @return probability
      */
     private static double getProbability(int red, int blue) {
           double max=0;
           int maxRed = 0;
           int maxBlue = 0;
          
           for (int i = 1; i < red; i++) {
                for (int j = 0; j < blue; j++) {
                     double prb = (0.5*i)/(i+j) + (0.5*(50-i))/(100-i-j);
                    
                     if (prb>max) {
                           max = prb;
                       maxRed=i;
                       maxBlue=j;
                     }
                }
           }
           System.out.println("Container 1, Red balls: "+maxRed +" Blue balls : "+maxBlue);
           System.out.println("Container 2, Red balls: "+(red-maxRed) +" Blue balls : "+(blue-maxBlue));
           return max;
     }
    
     /** Driver Method. */
     public static void main(String[] args) {
           int red = 50;
           int blue = 50;
           double p = getProbability(red,blue);
           System.out.println("Maximum probability of RED ball "+p);
     }
}
Output: 7474747474747475

                                                  

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