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